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A pictorial introduction to differential geometry,

leading to Maxwell’s equations as three pictures.

Dr Jonathan GratusPhysics Department, Lancaster University and the Cockcroft Institute of accelorator Science.

j.gratus@lancaster.ac.uk

http://www.lancaster.ac.uk/physics/about-us/people/jonathan-gratus

September 26, 2017

Abstract

In this article we present pictorially the foun-dation of differential geometry which is a cru-cial tool for multiple areas of physics, notablygeneral and special relativity, but also me-chanics, thermodynamics and solving differ-ential equations. As all the concepts are pre-sented as pictures, there are no equations inthis article. As such this article may be readby pre-university students who enjoy physics,mathematics and geometry. However it willalso greatly aid the intuition of an undergrad-uate and masters students, learning generalrelativity and similar courses. It concentrateson the tools needed to understand Maxwell’sequations thus leading to the goal of present-ing Maxwell’s equations as 3 pictures.

Prefix

When I was young, somewhere around 12, Iwas given a book on relativity, gravitationand cosmology. Being dyslexic I found read-ing the text torturous. However I really en-joyed the pictures. To me, even at that age,understanding spacetime diagrams was nat-ural. It was obvious, once it was explained,that a rocket ship could not travel more inspace than in time and hence more horizontalthan 45◦. Thus I could easily understand why,having entered a stationary, uncharged blackhole it was impossible to leave and that youwere doomed to reach the singularity. Like-wise for charged black holes you could escapeto another universe.

I hope that this document may give anyoneenthusiastic enough to get a feel for differen-tial geometry with only a minimal mathemat-ical or physics education. However it helpshaving a good imagination, to picture thingsin 3 dimension (and possibly 4 dimension)and a good supply of pipe cleaners.

I teach a masters course in differentialgeometry to physicists and this documentshould help them to get some intuition be-fore embarking on the heavy symbol bashing.

Contents

1 Introduction 3

2 Manifolds, scalar and vector fields 5

2.1 Manifolds. . . . . . . . . . . . . . . 5

2.2 Submanifolds. . . . . . . . . . . . . 6

2.3 Boundaries of submanifold . . . . . 6

2.4 Scalar fields. . . . . . . . . . . . . . 6

2.5 Vector fields. . . . . . . . . . . . . 6

3 Orientation. 8

3.1 Orientation on Manifolds (internal) 8

3.2 Orientation on submanifolds: In-ternal and external . . . . . . . . . 9

3.3 Internal and external, twisted anduntwisted . . . . . . . . . . . . . . 9

3.4 Concatenation of orientations . . . 10

3.5 Inheritance of orientation by theboundary of a submanifold . . . . 10

3.6 Orientations of vectors . . . . . . . 11

3.7 Twisting and untwisting . . . . . . 11

4 Closed forms. 11

1 File: Geometry-ArXiv.tex

4.1 Multiplication of vectors andforms by scalar fields. . . . . . . . 14

4.2 Addition of forms. . . . . . . . . . 14

4.3 The wedge product also known asthe exterior product. . . . . . . . . 14

4.4 Integration. . . . . . . . . . . . . . 15

4.5 Conservation Laws . . . . . . . . . 16

4.6 DeRham Cohomology. . . . . . . . 16

4.7 Four dimensions. . . . . . . . . . . 17

5 Non-closed forms. 18

5.1 The exterior differential operator. . 19

5.2 Integration and Stokes’s theoremfor non-closed forms. . . . . . . . . 19

6 Other operations with scalars, vec-

tors and forms 20

6.1 Combining a 1-form and a vectorfield to give a scalar field. . . . . . 20

6.2 Vectors acting on a scalar field. . . 20

6.3 Internal contractions. . . . . . . . . 20

7 The Metric 21

7.1 Representing the metric: Unit el-lipsoids and hyperboloids . . . . . 21

7.2 Length of a vector . . . . . . . . . 22

7.3 Orthogonal subspace . . . . . . . . 22

7.4 Angles and γ-factors . . . . . . . . 23

7.5 Magnitude of a form. . . . . . . . . 23

7.6 Metric dual. . . . . . . . . . . . . . 23

7.7 Hodge dual. . . . . . . . . . . . . . 24

7.8 The measure from a metric . . . . 24

8 Smooth maps 24

8.1 Pullbacks of p-forms. . . . . . . . . 25

8.2 Pullbacks of metrics. . . . . . . . . 27

8.3 Pushforwards of vectors. . . . . . . 27

9 Electromagnetism and Electrody-

namics 27

9.1 Electrostatics and Magnetostatics. 29

9.2 Maxwell’s equations as four space-time pictures . . . . . . . . . . . . 30

9.3 Lorentz force equation . . . . . . . 31

9.4 Constant space slice of Maxwell’sequations. . . . . . . . . . . . . . . 32

10 Conclusion 32

List of Figures

1 Zooming in on a sphere. . . . . . . 5

2 Zooming in on a corner. . . . . . . 5

3 0, 1, 2 and 3-dimensional manifolds. 6

4 A selection of submanifolds andtheir boundaries . . . . . . . . . . 7

5 Pathological examples which arenot submanifolds. . . . . . . . . . . 7

6 A representation of a scalar field. . 8

7 A scalar field as a function of x

and y. . . . . . . . . . . . . . . . . 8

8 A scalar field depicted as contours. 8

9 A representation of a vector field. . 8

10 Internal orientations. . . . . . . . . 8

11 Equivalent orientations. . . . . . . 9

12 Creating a helix. . . . . . . . . . . 10

13 Inheritance of internal orientationon a boundary. . . . . . . . . . . . 11

14 Inheritance of external orientationon a boundary. . . . . . . . . . . . 11

15 Representation of a top-form. . . . 12

16 An untwisted 2-form on a 3-dimmanifold. . . . . . . . . . . . . . . 12

17 An untwisted 2-form on a 3-dimmanifold. . . . . . . . . . . . . . . 12

18 An untwisted 1-form on a 3-dimmanifold . . . . . . . . . . . . . . . 12

19 Addition of 1-forms in 2-dim . . . 14

20 Wedge product of untwisted 1-form and 2-form. . . . . . . . . . . 14

21 Wedge product of two untwisted2-form. . . . . . . . . . . . . . . . . 14

22 Wedge product of three untwisted2-form. . . . . . . . . . . . . . . . . 15

23 Wedge product of twisted forms . . 15

24 Wedge product of a twisted andan untwisted form. . . . . . . . . . 15

25 Integration of a 2-form. . . . . . . 15

26 Integration of a 1-form. . . . . . . 16

27 A conservation law. . . . . . . . . . 16

28 Conservation law for all space. . . 17

29 DeRham Cohomology on an an-nulus. . . . . . . . . . . . . . . . . 17

30 DeRham Cohomology on a Torus. 17

31 Failed non-zero integrals. . . . . . 18

32 2-form in 4-dimensions: plane pic-ture. . . . . . . . . . . . . . . . . . 18

33 2-form in 4-dimensions: movingline picture. . . . . . . . . . . . . . 18

34 Example of a non-closed 1-form(black) in 2-dim. . . . . . . . . . . 19

35 A non-closed, non-integrable 1-form. 19

36 More non-closed forms. . . . . . . . 19

37 Demonstration of Stokes’s theorem. 20

38 Combining vectors and 1-forms . . 20

39 Demonstration of internal con-traction. . . . . . . . . . . . . . . . 20

2

40 Representations of the metric in3-dim space. . . . . . . . . . . . . . 21

41 The spacetime metric. . . . . . . . 21

42 The metric ellipsoid and length. . . 22

43 The metric hyperboloid and length. 22

44 The metric ellipsoid and orthogo-nality. . . . . . . . . . . . . . . . . 23

45 The metric hyperboloid and or-thogonality. . . . . . . . . . . . . . 23

46 The metric ellipsoid and angles. . . 23

47 The metric hyperboloid and γ-factor. . . . . . . . . . . . . . . . . 23

48 The metric, magnitude of a 1-form 24

49 The metric dual. . . . . . . . . . . 24

50 The Hodge dual. . . . . . . . . . . 25

51 The measure from the metric. . . . 25

52 Projection 3-dim to 2-dim. . . . . . 25

53 Projection 3-dim to 1-dim. . . . . . 26

54 Pullback, 2-dim to 3-dim, 2-form . 26

55 Pullback, 2-dim to 3-dim, 1-form . 26

56 Pullback, 1-dim to 3-dim, 1-form . 26

57 Pullback, tangential . . . . . . . . 26

58 Pullback, 3-dim to 2-dim, 2-form . 27

59 Pullback, 3-dim to 2-dim, 1-form . 27

60 Pullback, 3-dim to 1-dim, 1-form . 27

61 The pullback of a metric onto anembedding. . . . . . . . . . . . . . 27

62 The Push forward of the vector field. 28

63 The magnetic 2-form in space. . . 29

64 Gauss’s Law in electrostatics. . . . 29

65 The B and H fields from a wire. . 30

66 The B and H fields from a magnet. 30

67 Conservation of charge . . . . . . . 30

68 Conservation of magnetic flux . . . 31

69 The excitation fields . . . . . . . . 31

70 The constitutive relations in thevacuum . . . . . . . . . . . . . . . 31

71 The Lorentz force law. . . . . . . . 32

72 Faraday’s law of induction on theplane z = 0. . . . . . . . . . . . . . 32

73 Maxwell Amperes law on theplane z = 0. . . . . . . . . . . . . . 33

List of Tables

1 External orientations. . . . . . . . 9

2 Types of orientation for all objects. 10

3 Concatenation of orientations. . . . 10

4 Examples of twisting and untwist-ing. . . . . . . . . . . . . . . . . . . 11

5 The fields in electromagnetism . . 28

6 The relativistic field in electro-magnetism . . . . . . . . . . . . . . 28

1 Introduction

Differential geometry is a incredibly usefultool in both physics and mathematics. Dueto the heavy technical mathematics need todefine all the objects precisely it is not usuallyintroduced until the final years in undergrad-uate study. Physics undergraduates usuallyonly see it in the context of general relativ-ity, where spacetime is introduced. This isunfortunate as differential geometry can beapplied to so many objects in physics. Theseinclude:

• The 3-dimensional space of Newtonianphysics.

• The 4-dimensional spacetime of specialand general relativity.

• 2-dimensional shapes, like spheres andtorii.

• 6-dimensional phase space in Newto-nian mechanics.

• 7-dimensional phase-time space in gen-eral relativity.

• The state space in thermodynamics.

• The configuration space of physical sys-tem.

• The solution space in differential equa-tions.

There are many texts on differential geome-try, many of which use diagrams to illustratethe concepts they are trying to portray, forexample [1–8]. They probably date back toSchouten [8]. The novel approach adoptedhere is to present as much differential geom-etry as possible solely by using pictures. Assuch there are almost no equations in this ar-ticle. Therefore this document may be usedby first year undergraduates, or even keenschool students to gain some intuition of dif-ferential geometry. Students formally study-ing differential geometry may use this text inconjunction with a lecture course or standardtext book.

In geometry there is always a tension be-tween drawing pictures and manipulating al-gebra. Whereas the former can give you in-tuition and some simple results in low dimen-sions, only by expressing geometry in termsof mathematical symbols and manipulatingthem can one derive deeper results. In addi-tion, symbols do not care whether one is in 1,

3

3, 4 dimensions or even 101 dimensions (theone with the black and white spots), whereaspictures can only clearly represent objects in1, 2, or 3 dimensions. This is unfortunateas we live in a 4-dimensional universe (space-time) and one of the goals of this documentis to represent electrodynamics and Maxwell’sequations.

As well as presenting differential geometrywithout equations, this document is novel be-cause it concentrates on two aspects of dif-ferential geometry which students often finddifficult: Exterior differential forms and ori-entations. This is done even before introduc-ing the metric.

Exterior differential forms, which herein wewill simply call forms, are important bothin physics and mathematics. In electromag-netism they are the natural way to define theelectromagnetic fields. In mathematics theyare the principal objects that can be inte-grated. In this article we represent forms pic-torially using curves and surfaces.

Likewise orientations are important forboth integration and electromagnetism.There are two types of orientation, in-ternal and external, also called twistedand untwisted, and these are most clearlydemonstrated pictorially. Some physicistswill be familiar with the statement that themagnetic field is a twisted vector field, alsoknown as a pseudo-vectors or an axial vector.Here the twistedness is extended to formsand submanifolds.

The metric to late is one of the most im-portant objects in differential geometry. Infact without it some authors say you arenot in fact studying differential geometrybut instead another subject called differentialtopology. Without a metric you can think ofa manifold as made of a rubber, which is in-finity deformable, whereas with a metric themanifold is made of concrete. The metricgives you the length of vectors and the an-gle between two vectors. It can also tell youthe distance between two points. In higherdimensions the metric tells you the area, vol-ume or n-volume of your manifold and sub-manifolds within it. It gives you a measureso that you can integrate scalar fields.

General relativity is principally the study

of 4-dimensional manifolds with a spacetimemetric. It is the metric which defines curva-ture and therefore gravity. General relativitytherefore is the study of how the metric af-fects particles and fields on the manifold andhow these fields and particles effect the met-ric. There are many popular science booksand programs which introduce general rela-tivity by showing how light and other objectmove in a curved space, usually by showingballs moving on a curved surface. Thereforeone of the things we do not cover in this doc-ument are the spacetime diagrams in specialrelativity and the diagrams associated withcurvature, general relativity and gravity.

Since the metric is so important one mayask why one should study manifolds withoutmetrics. There are a number of reasons fordoing this. One is that many manifolds donot possess a metric. To be more precise,since on every manifold one may constructa metric, one should say it has no single pre-ferred metric, i.e. one with physical signif-icance. For example phase space does notpossess a metric, neither do the natural mani-folds for studying differential equations. Evenin general relativity, one may consider varyingthe metric in order to derive Einstein’s equa-tions and the stress-energy-momentum ten-sor. Transformation optics, by contrast, maybe considered as the study of spacetimes withtwo metrics, the real and the optical metric.As a result knowing about objects and oper-ations which do not require a metric is veryuseful. The other reason for introducing dif-ferential forms before the metric is that, ifone has a metric, then one can pass between1-forms and vectors. Thus they both lookvery similar and it is difficult for the studentto get an intuition about differential forms asdistinct from vectors.

One of the goals of this article is to in-troduce the equations of electromagnetismand electrodynamics pictorially. Electromag-netism lends itself to this approach as the ob-jects of electromagnetism are exterior differ-ential forms, and most of electromagnetismdoes not need a metric. Indeed the fourmacroscopic Maxwell’s equations can be writ-ten (and therefore drawn) without a metric.It is only when we wish to give the addi-

4

tional equations which prescribe vacuum orthe medium that we need the metric. Thishas lead to school of thought [9] which sug-gest that the metric may be a consequence ofelectromagnetism.

The challenge when depicting Maxwell’sequations pictorially is that they are equa-tions on four dimensional spacetime and welack 4-dimensional paper. Unfortunately,were we blessed with such 4-dimensional pa-per, we would have to live in a universewith 5 spatial dimensions and therefore 6-dimensional spacetime. Whatever field the5-dimensional equivalent of me would be at-tempting to represent in pictures he/she/itwould presumably be bemoaning the lack of6-dimensional paper.

This article is organised as follows. Westart in section 2 with basic differential ge-ometry, discussing manifolds, submanifolds,scalar and vector fields. In section 3 we in-troduce both untwisted and twisted orienta-tion, dealing first with orientations on sub-manifolds, since this is the easiest to trans-fer to other objects like forms and vectors.We then introduce closed exterior differentialforms, section 4, showing how one can addthem and take the wedge product. We thentalk about integration and hence conservationlaws. This leads to a discussion about man-ifolds with “holes” in them. Finally in thissection we discuss the challenge of represent-ing forms in four dimensions. In section 5we discuss non-closed forms and how to vi-sualise and integrate them. In section 6 wesummarise a number of additional operationone can do with vectors and forms. The met-ric is introduced in section 7 where we showhow we can use it to measure the length of avector, angles between vectors as well the themetric and Hodge dual. Section 8 is aboutsmooth maps which shows how forms can bepulled from one manifold to another. Finallywe reach our goal in section 9 which showshow to picture Maxwell’s equations and theLorentz force equation. We start with staticelectromagnetic fields as these are the easiestto visualise and then attempt to draw the fullfour dimensional pictures. We then return tothree dimensional pictures but this time one

(a) (b)

(c)

Figure 1: Zooming in on a sphere. As we gofrom (a) to (b) to (c), the patch of the spherebecomes flatter.

(a) (b)

Figure 2: Zooming in on a corner. As we gofrom (a) to (b), the patch does not get anyflatter. This is not a manifold.

time and two spatial dimensions. We thenconclude in section 10 with a discussion offurther pictures one may contemplate.

All figures are available in colour online.

2 Manifolds, scalar and

vector fields

2.1 Manifolds.

Manifolds are the fundamental object in dif-ferential geometry on which all other objectexist. These are particularly difficult to definerigorously and it usually take the best part ofan undergraduate mathematics lecture courseto define. This means that the useful toolsdiscussed here are relegated to the last fewlectures.

Intuitively we should think of a manifoldas a shape, like a sphere or torus, which issmooth. That is, it does not have any cor-ners. If we take a small patch of a sphereand enlarge it, then the result looks flat, likea small patch of the plane, figure 1. By con-trast if we take a small patch of a corner then,no mater how large we made it, it would stillnever be flat, figure 2.

5

(a) A 0-dimensionalmanifold: a point.

(b) A 1-dimensionalmanifold: a curve.

(c) A 2-dimensionalmanifold: a surface.

(d) A 3-dimensionalmanifold: a volume.

Figure 3: 0, 1, 2 and 3-dimensional mani-folds.

In figure 3 are sketched manifolds with 0,1, 2 and 3 dimensions. 0-dimensional mani-folds are simply points, 1-dimensional mani-folds are curves, 2-dimensional manifolds aresurfaces and 3-dimensional manifolds are vol-umes.

Topologically, there are only two types of 1dimensional manifolds, those which are linesand those which are loops. Higher dimen-sional manifolds are more interesting.

2.2 Submanifolds.

A submanifold is a manifold contained in-side a larger manifold, called the embeddingmanifold. The submanifold can be either thesame dimension or a lower dimension than theembedding manifold. Examples of embeddedsubmanifolds are given in figure 4

There are all kinds of bizarre pathologieswhich can take place when one tries to em-bed one manifold into another, such as selfintersection, figure 5a, or the a manifold ap-proaching itself figure 5b. However we willavoid these cases.

Submanifolds may either be compact ornon compact. A compact submanifold isbounded. By contrast a non compact sub-manifold extends all the way to “infinity” at

least in some direction. In figure 4 we see thatexamples 4a, 4c, 4f and 4g are all compact,whereas the others are not compact. A com-pact submanifold may either have a bound-ary, as in section 2.3 below, or close in onitself like a sphere or torus.

2.3 Boundaries of submanifold

Unlike the embedding manifold, it is also use-ful to consider submanifolds with boundaries.See figure 4. Both compact and non com-pact submanifolds can have boundaries. Theimportance of the boundary is when doingintegration, in particular the use of Stokes’theorem. The boundary of a submanifold isan intuitive concept. For example the bound-ary of a 3-dimensional ball is its surface, the2-dimensional sphere, figure 4f. We assumethat the boundary of an n-dimensional sub-manifold is an (n − 1)-dimensional subman-ifold. We observe that the boundary of theboundary vanishes.

2.4 Scalar fields.

Having defined manifolds, the simplest ob-ject that one may define on a manifold is ascalar field. This is simply a real numberassociated with every point. See figure 6.The smoothness corresponds to nearby pointshaving close values, also that these values canbe differentiated infinitely many times. Ascalar field on a 2-dimensional manifold maybe pictured as function as in figure 7. Alter-natively one can depict a scalar field in termsof its contours, figure 8. We will see below,section 4, that these contours actually depictthe 1-form which is the the exterior derivativeof the scalar field.

2.5 Vector fields.

A vector field is a concept most familiar topeople. A vector field consists of an arrow,with a base point (bottom of the stalk) ateach point on the manifold, see figure 9.Although we have only drawn a finite num-

ber of vectors, there is a vector at each point.The smoothness of the vector field corre-sponds to nearby vectors not differing too

6

(a) The interval(green) is a compact1-dim submanifoldof a 1-dim manifold.The boundary ofthe submanifold arethe two points, thisis a compact 0-dimsubmanifold (blue).

(b) The lower por-tion is a non com-pact 2-dim submanifold(green) of a 2-dim man-ifold. The boundaryof this submanifold isthe non compact 1-dimsubmanifold (blue).

(c) The disc (green) isa compact 2-dim sub-manifold of a 2-dimmanifold. The bound-ary of the disk is the1-dim circle (blue), acompact submanifold.

(d) The half plane isa non compact 2-dimsubmanifold (green) ofa 3-dim manifold. Itsboundary is the noncompact 1-dim curve(blue).

(e) A non compact 2-dim submanifold (blue)of a 3-dim manifold.It is the boundary ofthe non compact 3-dimsubmanifold consistingof it and all pointsabove it.

(f) A compact 2-dimsubmanifold, the sur-face of the sphere, ofa 3-dim manifold. Itis the boundary of thecompact 3-dim ball.

(g) A compact 2-dim sub-manifold, the surface of atorus. It is the boundaryof the compact 3-dim solidtorus.

Figure 4: A selection of submanifolds and their boundaries. Observe that the boundaries is itself asubmanifold, but it does not have a boundary. This implies ate the boundary must extend to infinityor close in on itself.

(a) A self intersectingcurve.

(b) A curve approach-ing itself.

Figure 5: Pathological examples which arenot submanifolds.

much.

To multiply a vector field by a scalar field,simply scale the length of each vector by thevalue of the scalar field at the base point ofthat vector. Of course if the scalar field isnegative we reverse the direction of the ar-row. We are all familiar with the addition ofvectors from school, via the use of a parallel-ogram.

The vector presented in figure 9 are un-twisted vectors. We will see below in section3.6 that there also exist twisted vectors.

7

1

2

4

0

2

-1

2

(a) Some of the valuesof the scalar field areshown.

1

2

4

0

2

-1

23

2

3

1

2

1

2

3

1

−1

2

1

5

2

1

(b) Intermediate val-ues of the scalar fieldare also shown.

Figure 6: A representation of a scalar field.

f(x,y)

x

y

Figure 7: A scalar field as a function of x

and y.

3 Orientation.

We now discuss the tricky role that orienta-tion has to play in differential geometry, es-pecially the difference between twisted anduntwisted orientations. However the distinc-tion is worth emphasising since the objects inelectromagnetism are necessarily twisted andothers are necessarily untwisted.

The take home message is:

x

y

Figure 8: A scalar field depicted as contours.The arrows point up.

Figure 9: A representation of a vector field.

+

−

(a) 0-dimmanifold.

(b) 1-dimmanifold.

(c) 2-dim mani-fold.

Right handed Left handed

(d) 3-dim manifold.

Figure 10: Possible (internal) orientations on0, 1, 2 and 3 dimensional manifold.

• Orientation is important for integrationand Stokes’s theorem.

• The orientation usually only affects theoverall sign. However it is important whenadding forms.

• The pictures are pretty.

3.1 Orientation on Manifolds(internal)

We start by showing the possible orientationson manifolds. Not all manifolds can have anorientation. If a manifold can have an ori-entation it is called orientable. Examples oforientable manifolds include the sphere andtorus. Non-orientable manifolds include theMobius strip and the Kline bottle. Even if a

8

Embedding dimension

0 1 2 3

0+ −

Subman

ifolddim

ension

1 + −

2+ −

3 + −

Table 1: Possible external orientations on a0, 1, 2 and 3-dim submanifold of a 0, 1, 2 and3-dim embedding manifold.

manifold is orientable, one may choose not toput an orientation on the it. For manifoldsof 0, 1, 2 or 3 dimensions the two possible(internal) orientations are shown in figure 10.These are as follows:

• An orientation for a 0-dimensional sub-manifold consists of either plus or minus,figure 10a. However convention usually dic-tates plus.

• On a 1-dimensional manifold it is anarrow pointing one way or another, figure10b.

• On a 2-dimensional manifold it is an arcpointing clockwise or anticlockwise, figure10c.

• On a three dimensional manifold oneconsiders a helix pointing one way or an-other, figure 10d. Interestingly, observethat unlike in the 1-dim and 2-dim cases,there is no need for an arrow on the helix.

3.2 Orientation on submani-folds: Internal and external

Again submanifolds may be orientable ornon-orientable. However with orientablemanifold one now has a choice as to whichtype of orientations to chose: internal or ex-ternal. With a manifold there is no conceptof an external orientation because there is noembedding manifold in which to put the ex-

Figure 11: Equivalent external orientationsof 1 and 2 dimensional submanifold in 3-dim.The actual length of the arc or arrow leavingis irrelevant. What matters is simply what thecirculation around 1-dim submanifold and sideof the 2-dim submanifold it points.

ternal orientation.The internal orientation of submanifolds

are exactly the same as those for manifolds,as shown in figure 10, with the type of the ori-entation: sign, arrow, arc or helix, dependingon the dimension of the submanifold.By contrast external orientations depend

not just on the dimension of the submanifoldbut also on the dimension of the embeddingmanifold. They are similar, plus or minussign, arrow, arc or helix, as the internal ori-entations. However in this case it takes placein the embedding external to the submani-fold. In table 1 we see examples of externalorientations for 0, 1, 2, 3 dimensional sub-manifolds embedded into 0, 1, 2, 3 dimen-sional manifolds. As we see, the type of theexternal orientation depends on the differencein the dimension of the embedding manifoldand submanifold. Thus the external orienta-tion is a plus or minus sign if the two mani-folds have the same dimension. It is an arrowif they differ by 1 dimension, an arc if theydiffer by 2 dimensions and a helix if they dif-fer by 3 dimensions.We emphasise that when specifying an ori-

entation, we can point the respective arrow orarcs in a variety of directions without chang-ing the orientation. For example, in figure11 we see how the same orientation can berepresented by several arcs or arrows.

3.3 Internal and external,

twisted and untwisted

For manifolds and submanifold it is useful tolabel internal orientations as untwisted and

9

Alwaysdefinable &Essential

Internal External

Manifold No Untwisted UndefinedSubmanifold No Untwisted TwistedVectors Yes Untwisted TwistedForms Yes Twisted Untwisted

Table 2: Objects which can have orientationsand the correspondence between untwistedversus twisted and internal versus external.

+followedby

becomeswhichis

followedby + becomes

whichis

followedby

becomeswhichis

followedby

becomeswhichis

followedby

becomeswhichis

followedby

becomeswhichis

Table 3: Concatenation of orientations.

external orientations as twisted. As we havestated, as well as manifolds and submani-folds, vectors and forms also have orienta-tions. These may also be twisted or untwistedand may also be internal or external. The pic-tures of the orientations depend on whetherthey are internal or external, but the assign-ment to which are twisted or untwisted de-pend on the object, according to table 2.

3.4 Concatenation of orienta-tions

We can combine two orientations together.Simply take the two orientations and jointhem together, the second onto the end of thefirst. See table 3.

You can see that the untwisted orientationsfor two dimensions, figure 10c, i.e. clockwiseor anticlockwise arise from the concatenation

z

x

y

(a) A curve created bygoing a short distancein x (red), followed bya short distance in z

(blue) followed by go-ing a short distance iny (green) and then re-peating.

z

yx

(b) The same diagramas in figure 12a but nowwith the axes all point-ing towards the reader.

z

yx

(c) The same diagram as in figure 12b butwith a offset each loop to show the helix.

Figure 12: Demonstration that going a shortdistance in x then z then y and repeating cre-ates a helix. The reader is encouraged to ex-periment with some pipe cleaners.

of two orientations, as in rows three and fourof table 3. Likewise the helices, figure 10darise by concatenating an arc with a perpen-dicular line as in rows five and six of table3.Using pipe cleaners may help you to see

that that an arc followed by a perpendicularline is equivalent to a helix. Likewise a helixarises in 3 dimensions by tracing out a path,going in one direction, then a second, thenthe third and repeating as in figure 12.

3.5 Inheritance of orientation

by the boundary of a sub-manifold

If a submanifold has an internal orientationand a boundary then the boundary inheritsan internal orientation, see figure 13. Like-

10

+

-

(a)Boundaryof a 1-dimsubmani-fold

(b)Boundaryof a 2-dimsubmani-fold

(c) Boundary of a3-dim submanifold.The helix is theorientation of the3-dim ball.

Figure 13: Inheritance of an internal orien-tation of a submanifold by its boundary for1,2,3-dimensional submanifold embedded intoa 3-dimensional manifold.

(a)Boundaryof a 1-dimsubmani-fold

(b)Boundaryof a 2-dimsubmani-fold

+

(c) Boundary of a3-dim submanifold,The + is the externalorientation of the3-dim ball and thearrow pointing in-wards is the externalorientation of the2-dim sphere.

Figure 14: Inheritance of an external orien-tation of a submanifold by its boundary for1,2,3-dimensional submanifold embedded intoa 3-dimensional manifold.

wise if a submanifold has an external orienta-tion then the boundary inherits the externalorientation, figure 14.

3.6 Orientations of vectors

In section 2.5 and figure 9 we saw the usualtype of vectors. These are untwisted vec-tors, since they have an untwisted orienta-tions which is the internal orientations. Thereexists another type of vector called a twistedvector. These are also called axial or pseudo-vectors. An example of such a vector is themagnetic fieldB. Twisted vectors have exter-

Embeddingdimension

Embeddingorientation

Submanifoldexternalorientation

Submanifoldinternalorientation

2

3

Table 4: Examples of twisting and untwist-ing.

nal orientations. They look like the externalorientations of 1-dimensional submanifolds asdepicted in table 1. In the case of vectors in3-dimension one can see why they are call ax-ial.

3.7 Twisting and untwisting

If a manifold has an orientation then we canuse it to change the type of orientation of asubmanifold. Thus we can convert an exter-nal orientation into an internal orientation.We can also convert an internal orientationinto an external orientation. The conventionwe use here is to say that if we concatenatethe external orientation (first) followed by theinternal orientation (second) one must arriveat the orientation of the manifold. See exam-ples in table 4.For example if the orientation of a 2-dim

manifold is clockwise and the untwisted ori-entation is up then the twisted orientation isright. See fourth line of table 3.Starting with an untwisted form, then

twisting and then untwisting it does notchange the orientation.

4 Closed forms.

In this section we introduce exterior differen-tial forms, which we call simply forms. On ann-dimensional manifold a form has a degree,which is an integer between 0 and n. We referfor a form with degree p as a p-form. A closedform of degree p can be thought of as a col-lection of submanifolds1 of dimension n − p.

1Some authors [2, 8, 9] use a double sheet to in-dicate 1-forms and likewise they replace 1-dim form-

11

⊕⊕ ⊕⊕

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(a) Low density.

⊕⊕⊕⊕⊕⊕⊕

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(b) Increased density,while reducing thevalue of each dot.

Figure 15: Representation of a twisted top-form, i.e. an n-form on an n-dimensionalmanifold.

Figure 16: An untwisted 2-form on a 3-dimmanifold.

submanifolds with cylinders to indicate (n−1)-forms.It is true that for a single 1-form at a point it isnecessary to have two (n − 1)-dim form-elements-submanifold in order indicate the magnitude of theform. Consider in figure 38 one needs at least 2-linesto indicate the combination of a 1-form with a vectorwhere both are defined only at one point. However inthis document we only consider smooth form-fields.I.e. a form defined at each point. Thus there is aform-submanifold passing through each point. Fur-thermore since the form-field is smooth form subman-ifolds for nearby points are nearly parallel. There-fore there is sufficient information to indicate themagnitude of the form via the density of the form-submanifolds. Looking at figure 38 again we see thereis sufficient information to calculate the action of thevector on the form.

Figure 17: An untwisted 2-form on a 3-dimmanifold.

Figure 18: An untwisted 1-form on a 3-dimmanifold. Note that the 1-form must vanishin the region between the two opposing orien-tations. A diagram of a twisted 1-form in 3dimensions is given in figure 26.

We call these form-submanifolds. Similarto a vector field, there is a submanifold of thecorrect dimension passing though each pointin the manifold. They must not have bound-aries, therefore they either continue off to “in-finity” or they close in on themselves, likespheres. In section 5 we see that the form-submanifolds with boundaries correspond tonon-closed forms.

Although the points, lines, surfaces corre-sponding to p-forms, for p ≥ 1, do not have

12

values, they do have an orientation and asbefore these can be untwisted or twisted. Asstated in table 2 these are the opposite wayround than for submanifolds. Thus an un-twisted form has an external orientation anda twisted form has an internal orientation.The untwisted forms are mathematically sim-pler and much easier to take wedge productsof. See section 4.3 below. Indeed the stan-dard method of prescribing the internal ori-entation of an n-dim manifold is in terms ofan untwisted n-form.

We have the following:• An untwisted 0-form (internal orienta-tion) is simply a scalar field and is bestvisualised as above in figures 6, 7 and 8.Note that a closed 0-form is a constant andnot very interesting. From the prescriptionabove, a 0-form on an n-dim manifold is ann-dim form-submanifold at each point.A twisted 0-form (external orientation)

requires an external orientation for a 0-dimsubmanifold (point) as in table 1.

• On an n-dimensional manifold, theform-submanifolds for an n-form consistsof a collection of dots. For the rest of thisreport, we shall refer to n-form as top-forms.For twisted top forms (internal orienta-

tion) these dots carry a plus or minus sign.See figure 15. The density of the dots cor-responds loosely to the magnitude of thetop-form. The smoothness of the field re-quires that between a high density of dotsand a low density dots, there should be aregion of intermediate density.A non-vanishing twisted top-form with

only plus dots is called a measure. Wesee below that these are very useful forintegrating scalar fields (section 4.4) andfor converting between vector fields and(n − 1)-forms (section 6.3). We see in sec-tion 7.8 that a metric automatically givesrise to a measure.Although, as with vector fields, there is

really a “dot at each point”, therefore thedensity of dots is actually infinite. Howeverwe can think of a limiting process. Startwith a set of discrete dots, then (approxi-mately) double the density of dots, while atthe same time halving the value associated

with each points. As such the number ofdots multiplied by their sign inside a par-ticular n-volume is approximately constantduring the limiting process. Compare thisto the scalar field, where as we stated, thevalue of each point remains constant duringthe limiting process.For untwisted top-forms each point car-

ries an external orientation, as in 0 dimen-sional submanifolds in table 1.

• On an n-dimensional manifold theclosed (n − 1)-forms correspond to 1-dimensional form-submanifolds, i.e. curvesas depicted in figures 16 and 17. The curvesdo not have start points or end points.Therefore they must either close on them-selves to form circles or they must go fromand to “infinity”. They also do not in-tersect. Again there is really a curve go-ing though each point. The smoothnessagain requires that between a high densityof curves and a low density there should bean intermediate density, and that betweencurves of opposite sign there should be agap.Figure 16 shows an untwisted 2-form (ex-

ternal orientation) in 3 dimensions.Twisted (n − 1)-forms are lines with di-

rection. (Internal orientation). One mayconsider these as flow lines of a fluid or sim-ilar. An example is shown in figure 17.You can see from figures 9 and 17 that

untwisted vectors and twisted (n−1)-formslook similar, and indeed they are. Withjust a measure we can convert a vector intoa 1-form via the internal contraction. Seesection 6.3.

• On an n-dimensional manifold theclosed 1-forms correspond to (n − 1)-dimensional form-submanifolds.Untwisted 1-forms are the submanifold

corresponding to the contours or level sur-faces of a scalar field, as in figure 8. Theexternal orientation point in the directionof increasing value of the scalar field. Thefaster the rate of increase of the scalar field,the closer the level surfaces.One usually takes the gradient of a scalar

field which is the vector corresponding tothe direction of steepest assent. This gra-dient is perpendicular to the level surfaces.

13

(a) Addition of untwisted 1-forms

(b) Additionof twisted1-forms

Figure 19: Addition of untwisted (lefttwo) and twisted (right) 1-forms in 2-dim.The solid 1-form-submanifold (blue and red)are added to produce the dashed 1-form-submanifold (green).

However to do this one needs a concept oforthogonality which comes with the metricdefined below in section 7.An example of an untwisted 1-form in 3-

dimensions is given in figure 18.

4.1 Multiplication of vectors

and forms by scalar fields.

To multiply a p-form field by a scalar fieldsimply increase the density of dots, curves,surfaces of the p-form by the value of thescalar field.

The action of multiplying a p-form by anegative scalar field changes the orientationof the p-form.

It is worth noting that if you multiply aclosed p-form with p < n by a non constantscalar the result will not be a closed form.

4.2 Addition of forms.

It is only possible to add forms of the samedegree and twistedness. Thus two untwistedp-forms sum to an untwisted p-form and twotwisted p-forms sum to a twisted p-form. Incontrast to multiplying by a scalar, addingtwo closed forms gives rise to a closed form.

Although algebraically adding forms istrivial, the visualisation of the addition of twop-forms is surprisingly complicated. Figure19 show how to add 1-forms in 2-dimension.

Figure 20: In 3-dimensions, the wedge prod-uct of an untwisted 1-form (blue) with a un-twisted 2-form (red) to give a untwisted 3-form (green).

Figure 21: In 3-dimensions, the wedge prod-uct of an untwisted 1-form (vertical, blue)with an untwisted 1-form (horizontal, red) togive an untwisted 2-form (green).

4.3 The wedge product alsoknown as the exterior prod-

uct.

In order to take the wedge product of twoforms simply requires taking their intersec-tion, figures 20-24. In places where the twoform-submanifolds are tangential then the re-sulting wedge product vanishes.

The wedge product of two untwisted formsis an untwisted form. Simply concatenate thetwo orientations. See figures 20 and 21. Thewedge product for three untwisted 1-forms isgiven by the intersection of three (n− 1)-dimform-submanifolds, figure 22.

The wedge product of two twisted forms isan untwisted form (figure 23) and the wedgeproduct of a twisted form and an untwistedform is a twisted form (figure 24). Howeverthe rules of this is more complicated thansimply concatenation. The only guaranteedmethod is to choose an orientation for the

14

Figure 22: In 3-dimensions, the wedge prod-uct of three untwisted 1-forms. The first verti-cal (blue) followed by the second facing reader(brown) then the third horizontal (red). Thisgives an untwisted 3-form (green).

Figure 23: In 3-dimensions, the wedge prod-uct of the twisted 1-form (vertical, blue) withthe twisted 1-form (horizontal, red) to givethe green untwisted 2-form.

manifold, then convert the twisted forms intountwisted forms, take the wedge product andthen apply a twisting if necessary. One cancheck that the result is invariant under thechoice of orientation of the manifold. How-ever it may depend on the twisting conventiondescribed in section 3.7.

4.4 Integration.

We can only integrate twisted top-forms, i.e.n-forms on an n-dimensional manifold, figure15. To do this we simply add the numberof positive dots and subtract the number ofnegative dots. Then take the limit the thedensity tends to infinity and the value tendsto zero. Let figure 15 refer to a 2-dimensionalmanifold. Thus in figure 15a, assuming eachdot value has 1, then its integral equals 9. In

Figure 24: In 3-dimensions, the wedge prod-uct of the untwisted 1-form (blue) with thetwisted 1-form (red) to give the green twisted2-form.

Figure 25: Integration of a twisted 2-form ona 2-sphere embedded in a 3-dimensional man-ifold. Observe that every curve that enters thesphere also leaves.

figure 15b each dot must have value 1

4and the

integral is 61

2. As the density of dots increases

and their value tends to zero, this sum willconverge. Contrast this to the case of thescalar field, figure 6, which does not convergeas the density increases. In order to integratea scalar field it is necessary to multiply it bya measure.

To integrate an untwisted top-form it isnecessary to use the orientation of the man-ifold to convert the untwisted top-form intoa twisted top-form. If the manifold is nonorientable, such as the Mobius strip then onecan only integrate twisted top-form.

What about integrating forms of lower di-mension? To integrate a p-form we must in-tegrate it over a p-dimensional submanifold.In addition untwisted forms are integratedover untwisted submanifolds and likewise for

15

Figure 26: Integration of a twisted 1-form ona twisted circle embedded in a 3-dimensionalmanifold. Observe that every surface thatcrosses the circle must cross again in the op-posite direction.

twisted forms and twisted submanifolds; withplus if the orientations agree and minus oth-erwise. We will see a generalisation of this insection 8.1 on pullbacks.

4.5 Conservation Laws

We can see from figures 25 and 26 that everycurve or surface that intersects the subman-ifold also leaves. Therefore the integral overthe sphere in figure 25 and over the circle infigure 26 are both zero. This gives rise toconservation laws which are very importantin physics. For example the conservation ofcharge or the conservation of energy. In fig-ure 27 we see that the closed twisted 3-form isintegrated over the surface of a 3 dimensionalcylinder. In this context we refer to the closed3-form as a current.

However there are three conditions on thesubmanifold which together imply that theintegral of a closed form is zero.

• The submanifold (of dimension p) mustbe the boundary of another submanifold (ofdimension p+ 1).

• The (p+ 1)-dimensional manifold mustnot go off to infinity.

• The larger manifold in which all thesubmanifolds are embedded must not haveany “holes”.

time

spacespace

space

initial time

final time

Figure 27: The closed twisted 3-form (red) isintegrated over the 3-dim surface of a cylinderwith net result zero. Any current which en-ters at the initial time (lower blue disc) musteither leave at the final time (upper blue disc)or through the curved surface of the cylinder(yellow). Note that the orientation of the sub-manifold is out (given by the yellow and bluearrows). This is because the sum of the inte-grals is zero. However the contribution frominitial time is negative.

See section 4.6 below for a comment aboutholes.In order to convert the integral of a closed

current into a conservation over all time itis necessary to expand the curved surface ofthe cylinder out to infinity. Thus the initialand final discs become all of space. One musttherefore guarantee that no current “escapesto infinity”. See figure 28.

4.6 DeRham Cohomology.

As mentioned in section 4.5 in order to guar-antee that the integral of a closed p-form overa closed p-dimensional submanifold is zero weneed that the larger manifold in which all thesubmanifolds are embedded has no “holes”.As an example of a manifold with holes con-sider the annulus, which is disc with a hole,given in figure 29. When integrating arounda circle which does not encircle the hole thenthe integral is zero. However when integrat-ing around the hole then the integral is non-zero.The use of integration to identify holes in a

16

initial time

final time

(a) A conserved field.

initial time

final time

(b) A non conserved field.

Figure 28: In 4-dimensions, integration ofthe twisted 3-form (red), showing it is con-served as long as nothing “escapes to infinity”.The 3-dimensional submanifolds (blue) corre-spond to “equal time slice”. Note that, com-paring with figure 27 we have let the curvedsurface (yellow) pass to infinity. We have alsoswapped the orientation of the initial time-slice so that it has the same orientation asthe finial time slice.

manifold is known as DeRham Cohomology.

Note that the “hole” need not be removedfrom the manifold like a hole is removed froma disc to create an annulus. In figure 30,one integrates over a circle which encircles the“hole” of a torus.

If the manifold does not have any holes,then as stated the integral is zero. It is inter-esting to see attempts to create a non zerointegral as in figure 31. In figure 31a theform-submanifolds terminate, whereas in fig-ure 31b the form-submanifolds all intersectat a point. In both cases these correspondto non closed forms as we will see below insection 5. The difference is that in 31a the 1-

Figure 29: A annulus which is a 2-dim man-ifold with a hole. When the untwisted 1-form(red) is integrated in a circle (blue) which doesnot enclose the hole the result is zero. By con-trast when one integrates over circle (green)which does enclose the hole the result is nonzero.

Figure 30: On a 2-dim torus, integrating the1-form (red) over the circle (blue) is not zero,despite the fact that the 1-form is closed.

form is continuous, in 31b it is discontinuous.We will see physical examples of such discon-tinuous forms in electrostatics and magneto-statics, section 9.1 which correspond to thefields generated by point and line sources suchas electrons and wires.

4.7 Four dimensions.

There are significant yet obvious prob-lems when trying to visualise forms in 4-dimensions. However there are importantreasons to try. As we will see below, we ofcourse live in a 4-dimensional universe andthe pictorial representation of electrodynam-ics would be particularly enlightening. Sec-ondly there are phenomena which only existin four or more dimensions. In lower dimen-

17

(a) In this case the form-submanifolds termi-nate. This corresponds to the 1-form (red) notbeing closed.

(b) In this case the form-submanifolds all inter-sect. This corresponds to the 1-form (red) notbeing continuous at this point.

Figure 31: Two attempts to create a nonzero integral as in figure 29, but on a man-ifold which does not have a hole. On the2-dimensional disc, the 1-form (red) is inte-grated over circle (green).

sions any p-form, for p ≥ 0 wedged with it-self is automatically zero. This is because ev-ery form-submanifold is tangential to itself.This does not apply to scalar fields, viewedas 0-forms, which can be squared. How-ever there are 2-forms in 4-dimensions whichare none zero when wedged with themselves.In particular, this includes the electromag-netic 2-form which we write F . Thirdly, allclosed form-submanifolds up to now have asingle smooth submanifold passing througheach point. However 2-forms in 4-dimensions,which are non-zero when wedged with them-selves, correspond, at each point, to two 2-

(a) t < 0 (b) t = 0 (c) t > 0

Figure 32: The plane representation of a2-form-submanifold in 4-dimensions. The 2-form-submanifold is represented by both theline (blue), for all time, and the momentaryplane (red) at the time t = 0. The self wedgeis represented the dot (green).

(a) t < 0 (b) t = 0 (c) t > 0

Figure 33: The moving line representation ofa 2-form-submanifold in 4-dimensions. The 2-form-submanifold is represented by two lines(red and blue). The self wedge is representedby the dot (green).

dimensional form-submanifolds which inter-sect at that point.One way to attempt to visualise a 2-form-

submanifold in 4-dimensions, is to use 3-dimensional space and 1-dimension of time.This is particularly relevant for visualisingspacetime. There are two possibilities. Let’sassume that the two 2-dimensional form-submanifolds intersect at time t = 0. Thenone possibility, shown in figure 32 is thatfor all t 6= 0 the form-submanifold is a line,whereas at t = 0 it is that line and an inter-secting plane. The alternative representationis given by two moving lines as seen in figure33. The result of wedging the 2-form withitself gives a 4-form. This top-form is repre-sented by a dot, which is the intersection ofthe 2-form-submanifold with itself.

5 Non-closed forms.

Non closed p-forms consist of a small sub-manifold of dimension (n − p) or surface at

18

Figure 34: Example of a non-closed 1-form(black) in 2-dim. If we integrate from bottomleft to top right on the green square, we seewe get a different value depending on the pathtaken.

Figure 35: An Attempt to show a non-closed,non-integrable 1-form in 3-dim (green). If wetry and close the elements up in to surfacewe see that we fail. Starting the red arc onthe lightest green and going round anticlock-wise we end up on the darkest green, we havehelixed down.

each point. We call these form-elements.The difference is that in non-closed forms, theform-elements do not connect up to form asubmanifold. Examples are given in figures 34and 25. We can see from figure 34 that theintegral now depends on path taken. Thusnon-closed forms correspond to non conser-vative fields.

In some (but not all) cases a non-closedform can be thought of as a collection ofclosed form-submanifolds, but each of thesehas a scalar field on it, see figure 36. Suchforms are known as integrable. For examplethe 1-form (in 2-dim) in figure 34 is integrablewhereas the 1-form (in 3-dim) in figure 35 isnot.

5.1 The exterior differential op-

erator.

The exterior differential operator takes a p-form and gives a (p+1)-form. One may thinkof this as taking the boundary of the form-submanifolds as given by figures 13 and 14.

Figure 36: An integrable non-closed 1-formin 3-dim: Two 1-form-submanifolds (green)each with a scalar field, represented by con-tour lines (red). The exterior differential ofthis 1-form are the closed red contour lines.We assume that the scalar field is increas-ing towards the inner contours. Both pos-sible twistedness orientations are shown onthe same diagram. For untwisted orientations(lower sheet, blue) concatenate the inward ar-row of the contours with the up arrow of thesubmanifold sheets. The twisted orientation(upper sheet, grey) simply inherits the orien-tation of the sheet.

The exterior derivative of a scalar field issimply the 1-form-submanifolds correspond-ing to the contours, also known as level (n−

1)-surfaces, of the scalar field. For untwistedscalar fields the orientation points in the di-rection the scalar field increases, as in figure8.

The exterior differential of an integrablenon-closed form are the contours of the scalarfield on each form-submanifold. See figure 36.

Clearly if the p-form is closed then it has noboundary and the result is zero. In addition,since a boundary has no boundary, the ac-tion of the exterior derivatives operator twicevanishes.

5.2 Integration and Stokes’s

theorem for non-closedforms.

The integration of non-closed forms is nolonger necessarily zero, even if there are noholes in the manifold. Recall to integratea p-form we must integrate it over a p-dimensional submanifold. In addition un-twisted forms are integrated over untwistedsubmanifolds an likewise for twisted forms

19

+

−

+

−+

−

+

−

+

−+

−

+

−

+

Figure 37: Demonstration of Stokes’s theo-rem in 2-dimensions. The orientation of thetwisted disk is + and the twisted circle is in-wards. The twisted 1-form is not closed andits exterior derivative is the positive and neg-ative dots. The integration of these dots overthe disk is −7. Likewise the integration of the1-form over the circle is also −7.

and twisted submanifolds; with plus if the ori-entations agree and minus otherwise.In figure 37 we see a demonstration of

Stokes’s theorem. That is, the integral of a(p−1)-form over a (p−1)-dimensional bound-ary of a submanifold equals the integral of theexterior derivative of the (p−1)-form over thep-dimensional submanifold. Again we have toassume the embedding manifold has no holes.With a bit of work defining the divergence

and curl in terms of exterior differential oper-ator, one can see that our version of Stokes’stheorem is a generalisation of the divergenceand Stokes’s theorem in 3-dimensional vectorcalculus.

6 Other operations with

scalars, vectors and

forms

6.1 Combining a 1-form and avector field to give a scalar

field.

Given a vector field and a 1-form field, onecan combine them together to create a scalarfield. The scalar is given by the number form-

Figure 38: Combining an untwisted 1-form(black) and an untwisted vector field (green)to give a scalar field, in 2-dim. The scalar atthe base point of the vector would have value−5.

Figure 39: In a 3-dim manifold. The inter-nal contraction of the untwisted 2-form (red)with the untwisted vector (blue) to producethe untwisted 1-form (green). The orienta-tion of the vector followed by the orientationof the 2-form gives the orientation of the 1-form.

submanifolds the vector crosses. See figure38. From this we see that if a 1-form con-tracted with a vector is zero, then the vectormust lie within the 1-form-submanifold.

6.2 Vectors acting on a scalar

field.

A vector acting on a scalar field gives a newscalar field. This is given by contracting theexterior derivative of the scalar field with thevector, as defined above. Thus it correspondsto the number of times a vector crosses thecontour submanifolds of the scalar field.

6.3 Internal contractions.

Internal contractions is an operation thattakes a vector field and a p-form and gives a(p−1)-form. Pictorially it corresponds to the(p − 1)-form-submanifold created by extend-ing the p-form-submanifold in the direction ofthe vector. Thus increases the dimension ofthe form-submanifold by 1. See figure 39. Ifthe vector lies in the p-form-submanifold theinternal contractions vanishes.

This is a generalisation of the contracting of

20

a vector and a 1-form given in subsection 6.1.Since the vector lies in the resulting (p − 1)-form-submanifold, internally contracting bythe same vector twice is zero.

7 The Metric

The metric gives you the length of vectors(figures 42 and 43) and the angle between twovectors (figures 44 and 45). It can also tellyou how long a curve is between to pointsand given two points it can tell you whichis the shortest such curve, which is called ageodesic. Thus it can tell you the distancebetween two points. This is the length alongthe geodesic between them.

In higher dimensions the metric tells youthe area, volume or n-volume of your mani-fold and submanifolds within it. It gives youa measure so that you can integrate scalarfields, figure 51. It enables you to convert 1-forms into vectors and vectors into 1-forms(called the metric dual). It also enables youto convert p-forms into (n − p)-forms (calledthe Hodge dual). It doesn’t, however, giveyou an orientation.

Metrics possess a signature. A list of n

positive or negative signs, where n is thedimension of the manifold. There are anumber of important dimensions and signa-tures:

• One dimensional Riemannian mani-folds.

• Two dimensional Riemannian mani-folds, i.e. signature (+,+). These are sur-faces, which may be closed like spheres andtorii, with a number of “holes”. They mayalso be open like planes. They also in-cluded non orientable surfaces such a theMobius strip.

• Three dimensional Riemannian mani-folds, i.e. signature (+,+,+). These in-clude the three dimensional space that welive in.

• Four dimensional spacetime, i.e. sig-nature (−,+,+,+). Here the minus signrefers to the time direction and the threepluses to space.

However for this document we will also belooking at three dimensional spacetime, sig-

(a) The unit spheres inflat Riemannian space.

(b) The unit spheres inRiemannian space.

Figure 40: Representations of the metric in3-dim space.

time

spacespace

(a) The two-sheet (upperand lower)hyperboloid.

spacetime

space

(b) The dou-ble cone.

spacetime

space

(c) The one-sheet hyper-boloid.

Figure 41: The spacetime metric (in 3 di-mensions). The two-sheet hyperboloid (a) isused for measuring timelike vectors. The up-per hyperboloid is for future pointing vectors,and the lower for past pointing vectors. Theone-sheet hyperboloid (c) is used for measur-ing the length of spacelike vectors. Betweenthe two hyperboloids is the double cone, whichall lightlike vectors lie on.

nature (−,+,+), and two dimensional space-time, signature (−,+). This is because theyare easier to draw.

7.1 Representing the metric:Unit ellipsoids and hyper-

boloids

Here we represent the metric is by the unit el-lipsoid or hyperboloid. See also [6]. Around apoint consider all the vectors which have unitlength. For a Riemannian manifold, thesevectors form an ellipse in 2 dimensions and anellipsoid in 3 dimensions. In figure 40 we showthe ellipsoid for a Riemannian manifold. Ifthe ellipsoids are the same at each point then

21

0.51

1.52

Figure 42: Using the metric ellipsoid to cal-culate the length of a vector. The unit ellip-soid is scaled to various sizes. The vector haslength 2.

the metric is flat as in figure 40a. By con-trast for a curved metric the ellipsoids mustbe different at each point as in figure 40b.

For spacetime metrics the ellipsoids is re-placed by a hyperboloid, as in figure 41. Sincethere is a minus in the signature, vectors aredivided into three types:

• Timelike vectors, these represent the 4-velocity of massive particles. These vectorspoint to one of the components of the two-sheet hyperboloid, as seen in figure 41a.As the name suggest, the two-sheet hyper-boloid has two components, the upper andlower. This distinguishes between futurepoint and past pointing vectors.

• Lightlike vectors, the 4-velocity of light.These vector lie on the double cone. See fig-ure 41b. Again one can distinguish betweenpast and future pointing lightlike vectors.

• Spacelike vectors, These are not the 4-velocity of anything. These vectors pointto the one-sheet hyperboloid. One cannotdistinguish between past and future space-like vectors.

7.2 Length of a vector

Using the metric ellipsoid or hyperboloid tocalculate the length of a vector is easy. Fora Riemannian metric the ellipsoid representall vectors of unit length. Therefore simplycompare length of the vector with the radiusof ellipsoid in the direction of the vector as in

1

2

time

1 2 space

Figure 43: Using the metric hyperboloid tomeasure the length of vectors.• The diagonal (green) lines are the path oflight rays. Thus the diagonal vector (darkgreen) is a lightlike vector.• The left and right (red) curves are the hyper-boloid for measuring the length of spatial vec-tors. (In two dimensions these two curves aredisjoint. However in three or more dimensionsthese two lines connect to create the one-sheethyperboloid.) The right-pointing (black) vec-tor is spatial and has length 2.• The upper and lower (blue) lines are for mea-suring the length of timelike vectors. The up-ward (orange) line is timelike and has length 1in the forward direction. The downward (pur-ple) line is timelike and has length 2 in thebackward direction.

figure 42.

We see in figure 43, we use the two-sheet(upper and lower) hyperboloid to measurethe length of forward and backward pointingtimelike vectors. We also use the one-sheethyperboloid (left and right curves) to mea-sure the spacelike vectors.

7.3 Orthogonal subspace

Given a vector then the metric can be used tospecify all the directions which are orthogonalto that vector. Orthogonal vectors are alsoknown as perpendicular vectors or vectors atright angles to each other. This is given bythe line tangent to the ellipsoid (figure 44) orhyperboloid (figure 45) at the point where thevector crosses it. In 3 dimensions the set oforthogonal directions form a plane and in 4dimensions they would form a 3-volume.

22

Figure 44: Using the metric ellipsoid to pre-scribe the orthogonal space (red) to a vector(blue). The orthogonal space is tangential tothe ellipsoid.

time

space

Figure 45: Using the metric hyperboloid toprescribe the orthogonal space (red) to a vec-tor (black). The orthogonal space is tangen-tial to the hyperboloid.

7.4 Angles and γ-factors

Given two vectors then a Riemannian metriccan be used to find the cosine of the anglebetween them as shown in figure 46. Thiscosine will always be a value between −1 and1.In the case of a spacetime metric, then in

figure 47 we get a value which is greater than1. We call this value the γ-factor. It is one ofthe key values in special relativity. Assumingthat the two vectors are the 4-velocity of twoparticles, then the γ-factor is a function of therelative velocity between the two particles.

7.5 Magnitude of a form.

One can use the metric to measure the mag-nitude of forms. On an n−dimensional Rie-mannian manifold the magnitude to a 1-form

ℓ

Figure 46: Using the metric ellipsoid to mea-sure the angle between two vectors. The(red) lines are orthogonal to one of the vec-tors (black). The length ℓ is the cosine of theangle between the two vectors.

time

space

γ

Figure 47: Using the metric hyperboloid tomeasure the angle between two vectors. The(red) lines are orthogonal to one of the vectors(black). The length γ is the γ-factor requiredto boost one of the vectors to the other.

is given by counting the (n − 1)-dimensionalform-submanifolds inside the ellipsoid and di-viding by 2. In figure 48 we see how touse the metric ellipse to measure the magni-tude to a 1-form in 2 dimensions. Measuringthe magnitude of a 2−form, requires count-ing the number of (n − 2)-dimensional form-submanifolds inside the ellipsoid and dividingby π. Similar formula exist for higher degreesand for spacetime manifolds.

7.6 Metric dual.

The metric can be used to convert a vec-tor into a 1-form and a 1-form into a vector.Given a vector then the 1-form is the (n−1)-form-submanifold which is perpendicular tothe vector, as given in figures 44 and 45. Seealso figure 49.

23

Figure 48: Using the metric to measure themagnitude of a 1-form in 2 dimensions. Sim-ply count the number of form-submanifold in-side the ellipse and divide by 2. In this casethe 1-form has magnitude 3.5

The twistedness of the orientation is un-changed. For spacelike vectors the orientationis preserved, whereas for timelike vectors theorientation is reversed. For lightlike vectors,the vector is orthogonal to itself. Thereforeit lies in its own orthogonal compliment. Forthese the orientation points in the directionof the nearby spacelike vectors.

7.7 Hodge dual.

The Hodge dual takes a p-form and gives theunique (n−p)-form which is orthogonal to it.It changes the twistedness of the form. In

Riemannian geometry, if one starts with anuntwisted p-form it gives a twisted (n − p)-form with the same orientation. If one startswith a twisted p-form it gives the untwistedform with either the same or the opposite ori-entation. This is given by double twisting asfollows:

• If p is even or (n − p) is even then theorientation is unchanged.

• If both p and (n − p) are odd then theorientation is reversed.

For non Riemannian geometry the formula forthe resulting orientation is more complicated.

7.8 The measure from a metric

A metric naturally gives rise to a measure.That is a twisted top-form. This is actu-ally the Hodge dual of the untwisted constantscalar field 1. It is given by packing togetherthe metric ellipsoids. See figure 51. We canintegrate this measure to give the size of anymanifold even of it is not orientable.

(a) Spacelike vector (blue). The orientation(red) of the 1-form is unchanged.

(b) Timelike vector (blue). The orientation(red) of the 1-form is reversed.

(c) Lightlike vector (blue). The orientation(red) of the 1-form points in the direction ofthe nearby spacelike vectors (green).

Figure 49: The metric dual converts a vectorinto a 1-form and visa versa. The untwistedvector (blue) is converted into the untwisted1-form (red).

8 Smooth maps

A smooth map is a function that maps onemanifold into another.

The simplest case of a smooth map is adiffeomorphism where there is a one to onecorrespondence between points in the sourcemanifold and points in the target manifold.As a result the source and target manifoldsmust have the same dimension and in somerespects they look the same, that is all theconcepts in sections 2-6. By contrast when ametric is included the two manifolds can lookvery different. One example is in transfor-mation optics where one maps between onemanifold where light rays travel in straightlines and another manifold where light raystravel around the hidden object.

One class of smooth maps that we have en-countered is that of the embedding of sub-manifolds as seen in figure 4. In this case wemap a lower dimensional (embedded) mani-fold into a higher dimensional (embedding)manifold. In this case every point in the em-

24

(a) In 2-dim. Hodge dual of the untwisted 1-form (red), becomes the twisted 1-form (blue).

(b) In 3-dim. Hodge dual of the untwisted 1-form (red), becomes the twisted 2-form (blue).

(c) In 3-dim. Hodge dual of the untwisted 2-form (red), becomes the twisted 1-form (blue).

Figure 50: Hodge dual of untwisted forms(red) to twisted forms (blue) in 2 and 3-dimensional Riemannian geometry. For space-time geometry the resulting orientation maybe reversed.

bedded submanifold corresponds to a uniquepoint in the embedding manifold. Howeversome points in the embedding manifold donot correspond to a point in the embeddedmanifold.

Another class are regular projections.These map a higher dimensional manifoldinto a lower dimensional manifold. See fig-ures 52 and 53. In this case every point in theimage manifold corresponds to a submanifoldof points in the domain.

Other maps however are none of the aboveand may include self intersections (figure 5),folding and a variety of other more compli-cated situations.

⊕

⊕

⊕

⊕

⊕

⊕

⊕

⊕

⊕

Figure 51: The measure from the metric. In2 dim, the twisted top form is given by packingtogether the metric ellipses.

Figure 52: A projection mapping a 3-dimmanifold onto a 2-dim manifold (green). Allthe points on each line (red and blue) are pro-jected onto the respective dots.

8.1 Pullbacks of p-forms.

No matter how complicated the map is it isalways possible to pullback the form. We willonly consider here the pullback with respectto diffeomorphisms, embeddings and projec-tions. Pullbacks preserve the degree of theform and also the twistedness of the forms.

The pullback with respect to diffeomor-phisms simply deforms the shape of the form-submanifolds. However unless there is a met-ric the question is: deforms with respect towhat? This corresponds to our statementthat without a metric one cannot distinguishbetween manifolds which are diffeomorphicand therefore as we said in the introduction,one is dealing with rubber sheet geometry.

For the pullback with respect to an em-bedding, one simply takes the intersectionof submanifold with the form-submanifoldsas seen in figures 54, 55 and 56. This pre-serves the degree of the form. If we pullbacka p-form from an m-dimensional manifold toan n-dimensional manifold then the p-form-

25

Figure 53: A projection mapping a 3-dimmanifold onto a 1-dim manifold (green). Allthe points on the plane (blue) are projectedonto the dot.

Figure 54: The pullback with respect to a 2-dim submanifold (red plane) embedded in a 3-dim manifold. The pullback of an untwisted 2-form (blue curves) gives an untwisted 2-form(green dot).

submanifolds are of dimension (m−p), hencethe intersection of the (m − p)-dim form-submanifolds with the n-dimensional mani-folds will have dimension (n− p) correspond-ing correctly to a p-form. This assumes theform-submanifolds and the submanifold in-tersects transversely. However if the subman-ifold and the form-submanifold intersect tan-gentially then the result is zero. See figure57.For untwisted forms there is a natural pull-

back of the orientation. For twisted forms it isnecessary to pullback onto a twisted subman-ifold. To be guaranteed to get the orientationcorrect it is easiest to choose an orientationfor the embedding manifold, untwist both theform and the submanifold and then twist thepullbacked form. This gives the result that:The orientation of the pullbacked form con-catenated with the orientation of the sub-manifold equals the orientation of the originalform. In particular the pullback of a twistedp-form onto a twisted submanifold is plus if

Figure 55: The pullback with respect to a 2-dim submanifold (red plane) embedded in a 3-dim manifold. The pullback of an untwisted 1-form (blue planes) gives an untwisted 1-form(green lines).

Figure 56: The pullback with respect to a1-dim submanifold (red curve) embedded in a3-dim manifold. The pullback of an untwisted1-form (blue plane) gives an untwisted 1-form(green dots).

the orientations agree and minus otherwise,as was discussed in section 5.2.

For the pullback with respect to projectionsthe resulting form-submanifold is simply allthe points which mapped onto the originalform-submanifold. See figures 58 and 59. Foruntwisted forms the orientation is obvious.For twisted forms one can use orientation forboth the source and target manifolds, untwistthe form and convert it back. Alternativelyone can decide an orientation of the projec-tion and use that. We will not describe this

Figure 57: The pullback with respect to a1-dim submanifold (red curve) embedded in a3-dim manifold. The 1-form (blue plane) istangential to 1-dim submanifold. The resultis zero.

26

Figure 58: The pullback with respect to aprojection from a 3-dim manifold to a 2-dimmanifold (green plane). The pullback of theuntwisted 2-form (blue dots) gives the 2-form(red lines).

Figure 59: The pullback with respect to aprojection from a 3-dim manifold to a 2-dimmanifold (green plane). The pullback of theuntwisted 1-form (blue curve) gives the 1-form(red plane).

here.

8.2 Pullbacks of metrics.

We can only pullback a metric with respectto diffeomorphisms and embedding. With re-spect to diffeomorphisms it makes the twomanifolds have the same geometry.

The pullback of a metric with respect toan embedding is called the induced metric.This is the metric we are familiar with on thesphere or torus which is embedded into flat3-space. Figure 61 shows the result of theinduced metric.

We cannot pullback a metric with respectto a projection as we don’t have enough in-formation. In particular we do not know thelength of vectors in the direction of the pro-

Figure 60: The pullback with respect to aprojection from a 3-dim manifold to a 1-dimmanifold (green curve). The pullback of theuntwisted 1-form (blue dot) gives the 1-form(red plane).

Figure 61: The pullback of a Riemannianmetric onto an embedding as represented byunit spheres. The unit spheres in 3-dim(green) are pulled back into unit circles (blue)on the 2-dim submanifold (red).

jection.

8.3 Pushforwards of vectors.

We can always pushforward a vector fieldwith respect to a diffeomorphism. However,in general the pushforward of a vector fieldwith respect to any other map will not resultin a vector field. As can be seen in figure 62some points in the target manifold have novector associated with them and others havemore than one.However the pushforward of vectors are

very useful. For example the velocity of asingle particle is the tangent to its worldline.This is only defined on its worldline.I will not discuss these further here.

9 Electromagnetism and

Electrodynamics

Maxwell’s equations, when written in stan-dard Gibbs-Heaviside notation are four dif-

27

Vectornotation

namesFormnotation

Relation betweenvector and form

Euntwistedvector

Electric field. Euntwisted1-form

Metric dual

Duntwistedvector

Displacement field. Dtwisted2-form

Metric dual andHodge dual

Btwistedvector

“B-field”, Magnetic field orMagnetic flux density.

Buntwisted2-form

Metric dual andHodge dual

Htwistedvector

“H-field”, Magnetic field intensity,Magnetic field strength,Magnetic field or Magnetising field.

Huntwisted1-form

Metric dual

ρuntwistedscalar

charge density. twisted3-form

Hodge dual

Juntwistedvector

current density. Jtwisted2-form

Metric dual andHodge dual

Table 5: The fields in electromagnetism, in terms of the familiar vector notation, the correspondingform notation and the operations needed to pass between the two.

Figure 62: Push forward of the vector field(red) of the 1-dim manifold (blue) embeddinginto the 2-dim manifold (white) produces vec-tors (red). However these vectors do not forma vector field as they are not defined in someplaces and in others have multiple values.

ferential equations of four vector fields E, B,D and H . The source for the electromag-netic fields is the charge density ρ and thecurrent J . In our notation it is clearer toreplace these with forms. In table 5 the rela-tionship between the vector notation and theform notation is given.As can be seen in table 5, both the fields

B and H may be referred to as the magneticfield. Here we will simply refer to them as the“B-field” and “H-field”.As we said, Maxwell’s equations are differ-

ential equations in E, B, D andH . Howeverthere is insufficient information to be able tosolve these equations. They need to be re-lated by additional information which relatethe four quantities E, B, D and H . Theseare called the constitutive relations. We have

Spacetimeform

Pull backonto space

Remainingcomponent

Funtwisted2-form

Buntwisted2-form

Euntwisted1-form

Htwisted2-form

Dtwisted2-form

Htwisted1-form

Jtwisted3-form

twisted3-form

Jtwisted2-form

Table 6: Combining the fields in electromag-netism into the relativistic quantities.

just stated the vacuum constitutive relations,namely that D = ǫ0E and B = µ0H . How-ever in a medium, such as glass, water, aplasma, a magnet, a human body or what-ever else you may wish to study, these con-stitutive relations are more complicated. Thesimplest constitutive relations consider a con-stant permittivity ǫ and permeability µ. Inthis medium there is no dispersion, i.e. ra-diation of different frequency travel at thesame speed. This would mean for examplethat rain drops would not give rise to rain-bows and may therefore be described as “an-tediluvian”! They are however easy to workwith. A more physically reasonable constitu-tive relations is to allow the permittivity orpermeability to depend on frequency. This isa good model for water, wax or the humanbody. If the electric or magnet fields are verystrong then the permittivity and permeability

28

depend on the electric or magnet fields. Theseare non linear materials. Exotic and meta-materials can have more complicated consti-tutive relations. The degree of complexity ofthese relations is limited only by the imagi-nation of the scientists studying the materialand their ability to build them. Indeed physi-cists are currently looking at models of thevacuum where the simple relations D = ǫ0E

and B = µ0H break down in the presentsof intense fields, either due to quantum ef-fects or because Maxwell’s equations them-selves breakdown.

Just as in relativistic mechanics where wecombine energy and momentum into a sin-gle relativistic vector the 4-momentum, sowe combine the electric and magnetic B-fieldinto a single spacetime untwisted 2-form F .Likewise we combine the displacement cur-rent and H-field into a single twisted 2-formH, and we combine ρ and J in a single twisted3-form J , as summarised in table 6. As a re-sult relativity mixes the E and B fields andseparately mixes the D and H fields. Theconstitutive relations then relate F and H.The simplest case is the vacuum where the re-lationship is simply given by the Hodge dual.

As stated, Maxwell’s equations give theelectric and magnetic fields for a given cur-rent. It is also necessary to know how thecharges, that produce the current, respond tothe electromagnetic fields. This is given bythe Lorentz force equation.

In this section we start, section 9.1, withelectrostatics and magnetostatics as these arein three dimensions and therefore easier todepict pictorially. We then, in section 9.2to draw Maxwell’s equations as four space-time pictures. In section 9.3, we depict theLorentz force equation and a spacetime pic-ture. In section 9.4 we look at Maxwell’sequations in 3-dimensional spacetime. This isthe pullback of the 4-dimensional spacetimeequations onto a fixed plane. This contraststhe the static cases when we pullback onto afixed timeslice.

Figure 63: The magnetic 2-form B in spaceas unbroken curves. Since the curves areunbroken this corresponds to the non diver-gence of B. The external orientation is given(green). The circle (red) encloses some of theB-lines.

D

E

Figure 64: Gauss’s Law in electrostatics. Thepositive charge, is given by the twisted 3-form is the dot (red) in the centre. This is theexterior derivative of the twisted 2-form D

(black) lines. In the vacuum or similar con-stitutive relations, then D is the Hodge dualof the untwisted 1-form E (green spheres).

9.1 Electrostatics and Magne-

tostatics.

In this subsection we deal with electrostat-ics and magnetostatics as 3-dimensional pic-tures are easier to draw. One has that B isdivergence-free, Gauss’s law which states thatthe divergence of D equals the charge, Fara-day’s law which states that E is curl-free andAmpere’s law which states that the curl ofH is the current. In the language of formsthis corresponds to the fact that B and E areclosed, whereas the exterior derivative of Dand H are ρ and J respectively.In figure 63 we see that the B lines are un-

broken and therefore B is closed. In figure 64we see Gauss’s law, that a charge twisted 3-form is the exterior derivative of the twisted2-form D. For the vacuum or for any consti-tutive relation where the permittivity is con-

29

Figure 65: Ampere’s law: The constant linecurrent, the twisted 2-form J (Black), is theexterior derivative of the twisted 1-form H. Inthe vacuum or similar constitutive relations,then H is the Hodge dual of the untwisted2-form B (blue circles).

stant or depends only on frequency, then theD is the permittivity times the Hodge dualof the electric 1-form E. In figure 65 we seeAmpere’s law. The current, a twisted 2-form

N

S

Figure 66: A permanent magnet. The un-twisted magnetic 2-form B (blue curves) andtwisted 1-form H (red ellipsoids) generated bya permanent magnet (grey rectangle). Ob-serve that outside of the magnet H fields areperpendicular to theB fields. Thus the Hodgedual, figure 50, of H coincides with B. Insidethe magnet however the orientations ofH andB are in opposite directions. This is due tothe non vacuum constitutive relations of per-manent magnets.

Tim

e

Space

Space

Space

Figure 67: Conservation of charge: Thereexists a closed twisted 3-form J called theelectric current (green). The green line maybe interpreted as charged particles, distributedover all of spacetime.

J in the wire, is the exterior derivative of thetwisted 1-form H , and H is the Hodge dualof B.

In figure 66 we show the fields for a per-manent magnet. This is an example of aslightly more complicated constitutive rela-tions. Outside of the magnet, the twisted 1-form H is the Hodge dual of the untwisted2-form B. However in the magnet, the ori-entations of H and B are in the opposite di-rection, due to the constitutive relations fora permanent magnet.

9.2 Maxwell’s equations as four

spacetime pictures

Figures 67 to 70 represent conservation ofcharge and Maxwell’s equations in spacetime.Thus there are four axes, one of time andthree of space.

We start with conservation of charge, fig-ure 67, which is not one of Maxwell’s equa-tions but a simple consequence of them. Infigure 67 we see the 1-dim form-submanifoldsof J do not have a boundary and thereforeJ is closed. One may think of the form-submanifolds of J as worldlines of the indi-vidual point charges such as electrons or ions.

Look again at figure 63, but now considerthat the fields are not static. We may imaginethat the magnetic field lines move in space.

30

Tim

e

Space

Space

Space

Figure 68: Conservation of magnetic flux:There exists a closed untwisted 2-form F

called the electromagnetic field (blue). Thisencodes the two fields E and B. Since we arein 4-dimensions the orientation is a loop out-side the 2-dim form-submanifolds. The clo-sure of F is equivalent to the two macroscopicMaxwell equations which contain E and B.

As a result some of the field lines may crossthe red circle. This will generate an electro-motive force as given by the integral form ofFaraday’s law of induction. In this case theform-submanifolds of B will map out a 2-dimform-submanifold in 4-dim spacetime. Theseform-submanifold will not have boundariesand therefore it corresponds to the untwistedclosed 2-form F . In figure 68 we depict theclosed 2-form F as 2-dim without boundary in4-dim spacetime. However one should recallthat 2-forms in 4-dimensions can self intersectas indicated in section 4.7.

Repeating the same procedure for figure 64.If the charges are not static then they mapout worldlines in spacetime. This is the closed3-form J as in figure 67. As a consequencethe 1-dim form-submanifolds for the 2-formD map out the 2-dim form-submanifolds forthe 2-form H. This gives figure 69.

As stated before, Maxwell’s equations needto be augmented using the constitutive rela-tions. In the vacuum one has that D and H

are the Hodge duals of E and B, as can beseen in figures 64 and 65. In figure 70 we seethat the twisted 2-form H is the Hodge dualof untwisted 2-form F . In other media therelationship between F and H is more com-

Tim

e

Space

Space

Space

Figure 69: The excitation fields: There ex-ists a twisted 2-form H called the excitationfield (red). It encodes the fields D and H. Ingeneral it is not closed and its exterior deriva-tive is J (green). This is equivalent to the twomacroscopic Maxwell equations which containD and H.

plicated.

9.3 Lorentz force equation

In electrodynamics there is one remainingequation, the Lorentz force law. As depictedin figure 71. This states that in spacetime the

Tim

e

Space

Space

Space

Figure 70: The constitutive relations in thevacuum: The twisted 2-form H (red) is theHodge dual of the untwisted 2-form F (blue).Here we attempt to show that H and F

are orthogonal 2-dim form-submanifolds in 4-dimensions. They have the same orientation.

31

Tim

e

Space

Space

Space

Figure 71: An attempt to draw the Lorentzforce law on a particle. Here the 2-form F isthe 2-dim form-submanifold (blue), the world-line of the particle is the curve (green) and theforce is the straight arrow (red). The force isorthogonal to to both the 2-form F and the4-velocity of the particle.

force is orthogonal to both the 4-velocity ofthe particle and the electromagnetic 2-formF .

9.4 Constant space slice ofMaxwell’s equations.

We may consider the static figures 63 and 64correspond to the pullback, as discussed insection 8, of the fields F , H and J on a 3-dimensional slice through spacetime.

An alternative slice of Maxwell’s equationsin spacetime is to choose a plane in space andconsider what the electromagnetic fields looklike when pulled back onto this plane for alltime. We will call this plane the z = 0 plane.The result is a diagram with one time coor-dinate and two space coordinates.

The 2-form F is a closed untwisted 1-dimensional form-submanifold. However thistime it contains information about both B

and E. Figure 72 shows Faraday’s law of in-duction for a loop in the plane z = 0. SinceF is closed the difference between the fieldat the top and bottom, which is the differ-ence in the B field, must correspond to fieldleaving the loop, which is the integral of theelectromotive force. Similarly figure 73 is theMaxwell-Amperes law.

time

space

Figure 72: Faraday’s law of induction on theplane z = 0, integrated over time and space.The blue lines are pullback of the electromag-netic 2-form F onto the plane z = 0. Theblue lines are continuous, i.e. as F is closedand therefore the pullback of F onto the planez = 0 is closed. Thus the difference betweenthe number of lines entering the bottom discand the number leaving the top disc equalsthe number of lines leaving the sides of thecylinder.

10 Conclusion

We have presented a large proportion of dif-ferential geometry, concentrating on differen-tial forms, in order to arrive at figures 67-71which are the diagrams that describe electro-dynamics.

In terms of pictorially presenting differen-tial geometry, there are still some open chal-lenges. One is in the attempt to depict 2-forms in 4 dimensions, section 4.7. One ques-tion is whether for a given 2-form, figure 32or 33 is correct and whether there are alter-natives which give more or different informa-tion.

There are many phenomena in electromag-netism which may gain from a pictorial rep-resentation. For example, how is it best todraw the the electromagnetism potential, andcan one shed light on the Aharonov–Bohmeffect? How does one depict the electromag-netic fields as well as the flow of electrons andions in a plasma. In this case, ideally onewould have 7 dimensional paper.

32

−

+

−

time

space

Figure 73: Maxwell Amperes law on theplane z = 0, integrated over time and space.The green dots are the intersections of thepositive and negative charges as they crossz = 0. The red lines are the pullback of theexcitation 2-form H onto the plane z = 0.These terminate at charges. Thus the differ-ence between the number of lines entering thebottom disc and the number leaving the topand sides of the cylinder equals the charge in-side the cylinder.

Acknowledgments

I am grateful grateful for the support pro-vided by STFC (the Cockcroft InstituteST/G008248/1 and ST/P002056/1), EPSRC(the Alpha-X project EP/J018171/1 andEP/N028694/1). I am grateful to the mas-ters students who took my course AdvancedElectromagnetism and Gravity and 2017 andread the document, making suggestions andto Taylor Boyd and Richard Dadhley whopointed out some inconsistencies.All figures are used with kind permission of

Jonathan Gratus [10].

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