chapter 1: vector analysis - universidad de sevilla · curso 2007/2008 dpto. física aplicada iii -...

Curso 2007/2008

Dpto. Física Aplicada III - Univ. de Sevilla

Joaquín Bernal Méndez 1

Chapter 1: Vector Analysis

Campos Electromagnéticos2º Curso Ingeniería IndustrialDpto.Física Aplicada III

Dpto. Física Aplicada III - Univ. de Sevilla 2

Chapter 1: Index (I)

Introduction

Scalar and Vector Fields

Integral CalculusCirculation

Flux

Differential CalculusThe Gradient

The Divergence

The Curl


Chapter 1: Index (II)

Mathematic toolsCylindrical and Spherical coordinates The “del” operatorThe Dirac delta function

Special types of fields:Curl-less or Irrotational fieldsDivergence-less or Solenoidal fieldsHarmonic fields

The Helmholtz Theorem


Introduction

Scalar: quantities characterized by its magnitude.Examples: weight, charge, temperature…

Vector: quantities that have direction as well as magnitude.Examples: velocity, acceleration, force…


Scalar Fields (I)

Definition: It is a function that associates a scalar value to every point in space.

Temperature: T(x,y,z)Geographical altitude: h(x,y)Density field of a body

A scalar field should give a single value of some variable for every point in space (monovalued function).


Scalar Fields (II)

Graphical representation: Equipotentialsurfaces ( , , )x y z Cϕ =

-2

-1

0

1

2 -2

-1

0

1

2

-2-1

0

1

-2

-1

0

1

-4 -2 0 2 4-4

-2

0

2

4


Scalar Fields (III)

Example: Atmospheric pressure

( , , )x y z Cϕ =


Vector Fields (I)

Definition: It is a function that associates a vector to every point in space.

Gravitational field

Velocity of a flowing liquid

Electric and magnetic fields

Monovalued functions


Vector Fields (II)Example: Wind field


Vector Fields (III)Graphical representation: A Field line is drawn such that the tangent to the field line at any point is parallel to the vector field at that point

x y z

dx dy dz

F F F= =


Vector Fields (IV)

Example: Electric field of a point charge

Positive charge


Vector Fields (V)

Example: Field lines of two point charges

Positive Charges Positive and negative charges

�-q

��q

�q

�q



Introduction



Flux


The Divergence

The Curl


Integral calculus

We have both scalar and vector fields

We can perform line, surface and volume integrals of these fields

From these, two types of integrals are important for us:

Circulation

Flux


Circulation (I): definitionLine integral of a vector field:

,

B

AF dr

γΓ = ⋅∫

It measures how much the vector field is aligned with the curve

The result of the integral is a scalar

The line integral depends on the path

Example: work performed by a force


Circulation (II): Physical meaningClosed path: indicates how much the vector field tends to circulate around the curve

·L

C F dl= ∫

0C =L

0C ≠L

For a force vector field this would be the work done in a closed path


Circulation: Further Examples

What is the sign of the circulation of these vector fields?


Circulation (III): calculation

Parametrize the curve:

Calculate the integral:

,

B

AF dr

γΓ = ⋅∫

{ }: ( ), t ( , )A Br r t t tγ = ∈

( ( ))B

A

t

t

drF r t dt

dtΓ = ⋅∫


Flux (I): definition

Surface integral of a vector field

The result is a scalar magnitude

It depends on the surface S

The direction of must be specified

For closed surfaces: is pointing outwards

SF dsΦ = ⋅∫

ss


Flux (II): Physical meaningThe flux of a vector field through a surface measures how much field crossed that surface

Example: velocity of a fluid

VSΦ =

·V SΦ =·

S

V dSΦ = ∫

d V dSΦ = ⋅


Flux (III): calculation

Parametrize the surface:

We have to calculate this double integral:

{ }1 2 1 2: ( , ), ( , ), ( , )S r r α β α α α β β β= ∈ ∈

r r

dS d dα βα β

⎛ ⎞∂ ∂= ×⎜ ⎟∂ ∂⎝ ⎠

( ( , )) r r

F r d dα β α βα β

⎛ ⎞∂ ∂Φ = ⋅ ×⎜ ⎟∂ ∂⎝ ⎠

∫∫


Summary

We will work with scalar and vector fieldsEquipotential surfaces are useful for representing scalar fieldsField lines are employed for representing vector fieldsCirculation of a vector field measures the “twist”of the vectorsThe flux of a vector field through a surface is proportional to the number of field lines passing through that surface



Introduction



Flux


The Divergence

The Curl


Differential calculus

The derivative of scalar and vector fields can be performed in different ways.

Scalar fields: gradient

Vector fields: divergence and curl


Gradient (I)For a function of one variable: ( )f x

0

0 0

0

( ) ( ) ( )lim

x

df x f x f x

dx ε

ε

ε→

+ −=

( )f x

xε

0( )f x ε+

0( )f x

The derivative tell us how rapidly the function varies when we change x


Gradient (II): directional derivative

¿How can we give an idea of the variation of a scalar function of several variables?

A direction must be specified:

Unit vector:

Directional derivative:

0

( , , ) ( , , )lim x y zx v y v z v x y zd

ds ε

ϕ ε ε ε ϕϕε→

+ + + −=

( , , )x y zv v v v=


Gradient (III)

By using the concept of partial derivative:

Definition:

Therefore:

x y z

dv v v

ds x y z

ϕ ϕ ϕ ϕ∂ ∂ ∂= + +∂ ∂ ∂

, , vx y z

ϕ ϕ ϕ⎛ ⎞∂ ∂ ∂= ⋅⎜ ⎟∂ ∂ ∂⎝ ⎠

grad x y zu u ux y z

ϕ ϕ ϕϕ ∂ ∂ ∂= + +∂ ∂ ∂

d=grad

dsv

ϕ ϕ ⋅


Gradient (IV): geometrical interpretation

gradϕ

v

α

The magnitude of the gradient at one point is the maximum value of the derivative at that point

The gradient vector points in the direction of maximun directional derivative (maximun increase of the function at that point)

v

gradϕ

0 cos 1α α= → =

d=grad

dsv⋅

ϕ ϕ

d= grad cos

ds

ϕ ϕ α


Gradient (V)

Increase of a given scalar function:

The gradient in a point is perpendicular to the equipotential surface at that point:

grad gradd v ds drϕ ϕ ϕ= ⋅ = ⋅

0 gradd drϕ ϕ= = ⋅

|grad drϕ −

Cteϕ =

drgradϕ


Gradient (VI): Summary

The Gradient is a vector field.Its magnitude gives the maximun value of the derivative of the function in that point.The gradient points in the direction of maximun rate of change of the scalar function.The gradient in a point is perpendicular to the equipotential surface at that point.



Introduction



Flux


The Divergence

The Curl


Divergence (I)

The divergence of a vector field is a scalar field:

Divergence in cartesian coordinates:

0

1div ( ) lim

S

F r F dSτ

τ τΔ

Δ →= ⋅

Δ ∫

div ( ) yx zFF F

F rx y z

∂∂ ∂= + +∂ ∂ ∂


Exercise

Calculate the divergence of the vector of position

div ( ) with ( )yx zFF F

F r F r rx y z

∂∂ ∂= + + =∂ ∂ ∂

) div 0

) div

) div 3

a r

b r x y z

c r

== + +=

Solution:


Divergence (II): Geometrical Interpretation

The divergence is a measure of how much the vector field spreads out (diverges) from a given point

Large positive divergence at P Zero divergence points

P


Divergence (III)

The Divergence Theorem:

vector pointing outward

Useful for evaluating integralsVery important for derivation of theoretical results

div ( )S

F r d F dSττ

τ = ⋅∫ ∫

z

xySτ

τ

dS


Curl (I)

Definition:

The Curl in cartesian coordinates:

0

1rot ( ) lim

S

F r dS Fτ

τ τΔ

Δ →= ×

Δ ∫

rot ( )

x y z

x y z

u u u

F r y zx

F F F

∂ ∂ ∂=∂ ∂∂

rot ( ) y yz x z xx y z

F FF F F FF r u u u

y z z x x y

∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞= − + − + −⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠


Curl (II): Geometrical Interpretation

The curls measures how much the vector field curls around a given point:

Nonzero curl Zero curl Zero curl


Stokes’ Theorem:

Right hand rule:If your finger point in the direction of the line integral then your thumb fixes the direction of

It is useful for:Calculation of integralsDerivation of important theoretical results

rotsS

F dS F drγ

⋅ = ⋅∫ ∫

Curl (III)

dS


Divergence and Curl: Summary

Derivatives of vector fields

Divergence: scalar field. Related with the existence of sources and sinks of the vector field

Rotacional: vector field. Related with the existence of “whirlpools” in the field lines

Fundamental Theorems:Divergence theorem

Stokes’ theorem



Introduction



Flux


The Divergence

The Curl







Curvilinear coordinates

We have seen several examples using cartesian coordinates

Many problems can be more easily solved by using other coordinates:

Cylindrical coordinates

Spherical coordinates

There are more coordinate systems but we will restrict ourselves to these.


Cartesian Coordinates (I)Any point is represented by three signed numbers, (x,y,z), where the coordinate is the perpendicular distance from the plane formed by the other two axesCoordinate lines: straight lines parallel to the axisCoordinate surfaces: planes parallel to the coordinate planes

X

Y

Z

r

xy

z z = cte

y = cte

x = cte

X

Y

Z

x

y

z

•P


Cartesian Coordinates (II)

Vector of position:

x y zr xu yu zu= + +

x y zdr dxu dyu dzu= + +

Orthogonal basis set:

Y

X

Z

r0

i j

k

uz

uy

ux

P

Differential elements of surface:

xy zdS dxdyu=

Differential volume elements:

zx ydS dxdzu=yz xdS dzdyu=

d dxdydzτ =

Infinitesimal displacement:


Cylindrical Coordinates (I)Three coordinates for a point P :

X

Y

Z

r

ρ

z

φ

ρ: perpendicular distance from the zaxisφ (azimuthal angle): angle around from the x axis

z (vertical c.): distance from the XY plane

0

0 2π

z

≤ < ∞≤ <

−∞ < < ∞

ρϕ

cos

sen

x

y

z z

===

ρ ϕρ ϕ ρ

φ

x

yX

Y

Z


Cylindrical Coordinates (II)

z=cte

z

ϕ=cte

ϕ

ρ

ρ=cteX

Y

Z

•P

Coordinate lines:ρ: horizontal straight half-linesφ: Horizontal circumferencesz: Vertical straight lines

Coordinate surfaces:ρ=cte.: vertical cylindersφ=cte: vertical half-planesz=cte: horizontal planes


Cylindrical coordinates (III)

Orthogonal basis set

Y

X

Z

r0

ρ

uρ

ux uy

uz

φ uφ

z

uz

P

cos senx y zr u u zuρ ϕ ρ ϕ= + +Vector of position:

zr u zuρρ= +

zdr d u d u dzuρ ϕρ ρ ϕ= + +

The unit vectors change direction as P moves around

Infinitesimal displacement:


Cylindrical coordinates (IV)

z=cte

z

ϕ=cte

ϕ

ρ

ρ=cteX

Y

Z

•P

Elements of surface:

Element of volume:d d dzdτ ρ ρ ϕ=

cte : dS d dzuρρ ρ ϕ= =

cte : zz dS d d uρ ϕ ρ= =

cte : dS dzd uϕϕ ρ= =

Sometimes letter r is used in instead of ρ


Gradient, divergence and Curl in Cylindrical Coordinates

1grad z

f f ff u u u

z

∂ ∂ ∂= + +∂ ∂ ∂ρ ϕρ ρ ϕ

( )1 1div z

F F FF

z

∂ ∂ ∂= + +

∂ ∂ ∂ρ ϕρ

ρ ρ ρ ϕ

1 1rot ( )z z

z

F F FF FF u u F u

z z

∂ ∂ ∂⎡ ⎤ ⎡ ⎤ ⎡ ⎤∂ ∂ ∂= − + − + −⎢ ⎥ ⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦ ⎣ ⎦

ϕ ρ ρρ ϕ ϕρ

ρ ϕ ρ ρ ρ ϕ


Spherical Coordinates (I)

X

Y

Z

r

φ

θ r

r (radial): distance from the origin

θ (polar): angle down from the positive z axis

φ (azimuthal): angle from the positive x-axis to the orthogonal projection of the position vector in the XY plane

0

0 π

0 2π

r≤ < ∞≤ ≤≤ <θϕ

sen cos

sen sen

cos

x r

y r

z r

θ ϕθ ϕθ

===

ρ

φ

x

yX

Y

Z

ρ

θ z

Z

r


Coordinate lines:r: radial half-lines from the origin

φ: horizontal circumferences (parallels)

θ: vertical circumferences (meridians)

Coordinate surfaces:r=constant: Concentric spheres

φ=constant: Vertical half-planes

θ=constant: Cones

Spherical Coordinates (II)

ϕ=cte ϕ

θ=cte

θr=cte

r

X

Y

Z

•P


Spherical Coordinates (III)

Orthogonal basis set

Y

X

Z

r0

ρ

ux uy

uz

ur

φ uφ

z

P

uθθ

sen cos sen sen cosx y zr r u r u r uθ ϕ θ ϕ θ= + +

Vector of position:

These unit vectors change direction as Pmoves around

rr ru=

senrdr dr u rd u r d uθ ϕθ θ ϕ= + +

Line element:


Spherical Coordinates (IV)

ϕ=cte ϕ

θ=cte

θr=cte

r

X

Y

Z

•P

Surface elements:

Volume elements:2 send r drd dτ θ θ ϕ=

2cte : sen rr dS r d d uθ ϕ θ= =

cte : sendS r d druθθ θ ϕ= =

cte : dS rd druϕϕ θ= =


Gradient, Divergence and Curl in Spherical Coordinates

1 1grad

senr

f f ff u u u

r r r

∂ ∂ ∂= + +∂ ∂ ∂θ ϕθ θ ϕ

22

1 1 1div ( ) (sen )

sen senr

FF r F F

r r r r

∂∂ ∂= + +

∂ ∂ ∂ϕ

θθθ θ θ ϕ

(sen ) ( )1 1 1rot

sen sen

1 ( )

rr

r

F rFF FF u u

r r r

rF Fu

r r

ϕ ϕθθ

θϕ

∂ θ ∂⎡ ⎤ ⎡ ⎤∂ ∂= − + − +⎢ ⎥ ⎢ ⎥θ ∂θ ∂ϕ θ ∂ϕ ∂⎣ ⎦ ⎣ ⎦

∂ ∂⎡ ⎤−⎢ ⎥∂ ∂θ⎣ ⎦







The Operator “ (I)

It allows a shorthand notation

Definition:

Vector operator :It acts upon (differentiate) the function to the right

It behaves like an ordinary vector

x y zu u ux y z

∂ ∂ ∂∇ = + +

∂ ∂ ∂


The operator “ (II)

grad

div

x y z

yx z

u u ux y z

FF FF F

x y z

ϕ ϕ ϕϕ ϕ∂ ∂ ∂= + + = ∇∂ ∂ ∂

∂∂ ∂= + + = ∇ ⋅∂ ∂ ∂

rot

zyx

yx z

uuu

F Fy zxF F F

∂ ∂ ∂= = ∇×∂ ∂∂


The operator “ (III)

“ can be expressed in any system of coordinates.Calculus carried out with the help of “ are independent of the coordinate system.Any identity that can be proved by using the cartesian coordinates version of “remains valid for any other system of coordinates.


Product Rules (I)

A scalar field can be obtained as the product of two other fields:

Product of two scalar fields:

Dot product of two vector fields:

What is the gradient of the product?

ψϕ

F G⋅

( )ϕψ ϕ ψ ψ ϕ∇ = ∇ + ∇

( ) ( ) ( ) ( ) ( )F G F G F G G F G F∇ ⋅ = × ∇× + ⋅∇ + × ∇× + ⋅∇

x y zF F F Fx y z

∂ ∂ ∂⋅∇ = + +

∂ ∂ ∂ ( ) ( )F G F G⋅∇ ≠ ∇ ⋅


Product Rules (II)Also, a vector field can be obtained from a product of fields:

Scalar and vector fields:Cross product of two vector fields: F G×

Fϕ

( )( )F F Fϕ ϕ ϕ∇ ⋅ = ∇⋅ + ∇ ⋅

( ) ( )F F Fϕ ϕ ϕ∇× = ∇× + ∇ ×

( ) ( ) ( )F G F G F G∇⋅ × = ∇× ⋅ − ⋅ ∇×

( ) ( ) ( ) ( ) ( )F G F G F G G F G F∇× × = ∇ ⋅ − ⋅∇ − ∇ ⋅ + ⋅∇

Divergence:

Curl:


Product Rules : Summary

( )ϕψ ϕ ψ ψ ϕ∇ = ∇ + ∇

( )( )F F Fϕ ϕ ϕ∇ ⋅ = ∇⋅ + ∇ ⋅

( ) ( )F F Fϕ ϕ ϕ∇× = ∇× + ∇ ×

( ) ( ) ( )F G F G F G∇⋅ × = ∇× ⋅ − ⋅ ∇×

( ) ( ) ( ) ( ) ( )F G F G F G G F G F∇ ⋅ = × ∇× + ⋅∇ + × ∇× + ⋅∇

( ) ( ) ( ) ( ) ( )F G F G F G G F G F∇× × = ∇ ⋅ − ⋅∇ − ∇ ⋅ + ⋅∇

Gradient:

Divergence:

Curl:


Second derivatives (I)By applying ∇ twice we can construct fivespecies of second derivatives:

The gradient is a vector field:Divergence of gradientCurl of gradient

The divergence is a scalar field:Gradient of divergence

The curl is a vector field:Divergence of curlCurl of curl

( )ϕ∇ ⋅ ∇( )ϕ∇× ∇

( )F∇⋅ ∇×( )F∇× ∇×

( )F∇ ∇⋅


Second derivatives (II)

2 2 22

2 2 2( )

x y z

ϕ ϕ ϕϕ ϕ∂ ∂ ∂∇ ⋅ ∇ = + + = ∇

∂ ∂ ∂( ) 0ϕ∇× ∇ =

( ) 0F∇⋅ ∇× =

2( ) ( )F F F∇× ∇× = ∇ ∇ ⋅ −∇

( )F∇ ∇⋅

Laplacian2( )∇

Seldom occurs2¡ ( ) ( ) !F F F∇ ∇⋅ ≠ ∇ ⋅∇ = ∇

Very important

Very important

Already defined







The Dirac Delta Function (I)

Consider this vector field:

Radial and pointing outwards, but:

However, by integrating over a sphere (R):

2 3ru r

vr r

= =

22 2

1 10v r

r r r

∂ ⎛ ⎞∇ ⋅ = =⎜ ⎟∂ ⎝ ⎠

2 220 0

sen 4rr

S

uv d v dS u R d d

Rτ

π π

τ

τ θ θ ϕ π∇⋅ = ⋅ = ⋅ =∫ ∫ ∫ ∫Divergence theorem


The Dirac Delta Function (II)The source of the problem is the point

Summing up, the function fulfills:

We have found a “weird” function: the Dirac delta function

0r =

22 2

0

1 1¡¡ !!

r

v rr r r =

∂ ⎛ ⎞∇ ⋅ = = ∞⎜ ⎟∂ ⎝ ⎠

2ru

r∇⋅

2 2

0 0 con 4

0r r

ru ud

rr rτ

τ π≠⎧

∇ ⋅ = ∇ ⋅ =⎨∞ =⎩∫


The Dirac Delta Function (III)The one-dimensional Dirac delta function:

Distribution: the limit of a sequence of functions

-

0 0( ) con ( ) 1

0

xx x dx

xδ δ

∞

∞

≠⎧= =⎨∞ =⎩

∫

2

2

0 0

1( ) lim ( ) lim e

π

x

x x−ε

εε→ ε→δ = δ =

ε


The Dirac Delta Function (IV)

0

1

2

3

4

5

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

δε(x

)

x

ε=1ε=0.5ε=0.25

ε=0.125

2

2εε

1( ) e

ε π

x

xδ−

=


The Dirac Delta Function (IV)Product with an ordinary function:

It is sufficient that the domain extend across the delta function:

The “spike” can be shifted:

( ) ( ) ( ) (0) ( ) ( ) (0)x f x x f x f x dx f∞

−∞

= ⇒ =∫δ δ δ

( ) ( ) (0)x f x dx f−

=∫ε

ε

δ

( ) ( ) ( )x a f x dx f a∞

−∞

− =∫ δ


z

x

y

τ

The Dirac Delta Function (V)The three-dimensional delta function:

In general:

3( ) ( ) ( ) ( )r x y zδ δ δ δ=

( )( ) ( )

0

a ar r a d

aτ

ϕ τϕ δ τ

τ∈⎧

− = ⎨ ∉⎩∫

3( ) ( ) ( ) ( )x y zr a x a y a z aδ δ δ δ− = − − −

a

a


The Dirac Delta Function (VI)

Coming back to the function

It can be written as:

2ru

r∇⋅

2 2

0 0 with 4

0r r

ru ud

rr rτ

τ π≠⎧

∇ ⋅ = ∇ ⋅ =⎨∞ =⎩∫

24 ( )ru

rr

πδ∇ ⋅ =

003

0

4 ( )r r

r rr r

πδ⎛ ⎞−

∇ ⋅ = −⎜ ⎟⎜ ⎟−⎝ ⎠0

30 0

1 r r

r r r r

⎛ ⎞ −∇ = −⎜ ⎟− −⎝ ⎠2

00

14 ( )r r

r rπδ

⎛ ⎞∇ = − −⎜ ⎟−⎝ ⎠







Curl-less vector fields:

Equivalent conditions:

There exits a scalar field such that:

Irrotational Fields

0F∇× =

0F drγ

⋅ =∫

1 2, ,

B B

A A

F dr F drγ γ

⋅ = ⋅∫ ∫

F = −∇ϕA

B

1γ2γ

( ) 0S

F dr F dSγγ

⋅ = ∇× ⋅ =∫ ∫


Divergence-less vector fields:Propiedades:

Flux is constant through a field tube:

There exists a vector field such that:

Solenoidal Fields0F∇⋅ =

0S

F dSτ

⋅ =∫

1 2

1 2

si s s

S S

F dS F dS⋅ = ⋅ γ = γ∫ ∫

F A= ∇×

0S

F dS F dτ τ

⋅ = ∇⋅ τ =∫ ∫

S1

S2

SL

2dS

1dS1 2s sγ = γ


Types of Vector Fields

0

0

F

F

∇⋅ ≠

∇× ≠

Irrotational

Solenoidal

Solenoidal andirrotational

0

0

F

F

∇⋅ ≠

∇× =

0

0

F

F

∇⋅ =

∇× ≠

0

0

F

F

∇⋅ =

∇× =


Harmonic Fields

Scalar fields satisfying:

Example: consider a vector field which is irrotational and solenoidal:

2 0∇ ϕ = Laplace equation

0F∇× = F⇒ = −∇ϕ

0F∇⋅ = ( ) 0⇒ ∇⋅ −∇ϕ = 2 0⇒ ∇ ϕ =

Practical case: electrostatic field in a region without charges








Given we can calculate: and

Given and Is it possible to get ?

Let:Scalar sources

Vector sources

If this is insufficient information: many solutions

If this is too much information: no solution

F∇⋅ F∇×F

FF∇⋅ F∇×

F∇⋅ = ρF c∇× = ( 0)c∇⋅ =


Helmholtz theorem: statement

The system with defined in all the space with:

has a single solution given by:

with:

;F∇⋅ = ρ F c∇× = 0c∇⋅ =

2 2lim ( ) 0 ; lim ( ) 0 ; lim ( ) 0r r r

r r r c r F r→∞ →∞ →∞

ρ = = =

F A= −∇ϕ+∇×

1 1

1

1 ( )( ) y

4 esp

r dr

r r

ρ τϕ =

π −∫ 1 1

1

1 ( )( )

4 esp

c r dA r

r r

τ=

π −∫Scalar potential Vector potential

Field pointSource point



Introduction



Flux


The Divergence

The Curl

chapter 1: vector analysis - universidad de sevilla · curso 2007/2008 dpto. física aplicada iii -...

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