vector calculus - minimal preparation course for …differentiation integration vector calculus...
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DifferentiationIntegration
Vector CalculusMinimal preparation course for 1st year electromagnetism
Shinsuke Kawai
Department of Physics/University College, Sungkyunkwan University
Autumn semester 2010
Shinsuke Kawai Vector Calculus
DifferentiationIntegration
Outline
1 DifferentiationDifferential operatorsGradient, divergence, rotation
2 IntegrationIntegrations in vector calculusIntegration formulaeMaxwell’s equations
Shinsuke Kawai Vector Calculus
DifferentiationIntegration
Differential operatorsGradient, divergence, rotation
Differential operators
Physical observables we have studied are scalars (such asenergy, charge density, mass density, etc.) and vectors (velocity,electric field, magnetic field, etc.). There are also axial vectors,tensors, etc.
They are functions of positions (x , y , z), as well as of time t .
Let us forget about the time dependence now.
Differentiation by one variable (say, x) while treating others (y andz) as constants – partial differentiation ∂
∂x
Let us make a vector from the differential operators ∂∂x
, ∂∂y
, and∂
∂z
:
~— = (∂
∂x
,∂
∂y
,∂
∂z
). (1)
Shinsuke Kawai Vector Calculus
DifferentiationIntegration
Differential operatorsGradient, divergence, rotation
Products of vectors
For vectors~A = (Ax
,Ay
,Az
) and~B = (Bx
,By
,Bz
), we can define
the scalar product: ~A ·~B = A
x
B
x
+A
y
B
y
+A
z
B
z
.
the vector product:
~A⇥~
B =
������
~ı ~j
~k
A
x
A
y
A
z
B
x
B
y
B
z
������
= (Ay
B
z
�A
z
B
y
,Az
B
x
�A
x
B
z
,Ax
B
y
�A
y
B
x
).
Shinsuke Kawai Vector Calculus
DifferentiationIntegration
Differential operatorsGradient, divergence, rotation
Operation with ~—
On a scalar f (x ,y ,z),
Gradient:
grad f = ~—f = (∂ f
∂x
,∂ f
∂y
,∂ f
∂z
).
On a vector ~F(x ,y ,z),
Divergence: div
~F = ~— ·~F = ∂F
x
∂x
+ ∂F
y
∂y
+ ∂F
z
∂z
.
Rotation (or curl):
rot
~F = curl
~F = ~—⇥~
F = (∂F
z
∂y
� ∂F
y
∂z
,∂F
x
∂z
� ∂F
z
∂x
,∂F
y
∂x
� ∂F
x
∂y
).
Note: div is a scalar, whereas grad, rot are vectors.
Shinsuke Kawai Vector Calculus
DifferentiationIntegration
Integrations in vector calculusIntegration formulaeMaxwell’s equations
Integration
In 3 dimensions we may consider:
Line integration: along d
~s
Surface integration: over an area vector d
~A
(|dA| is the area and the direction is perpendicular to the surface.Pointing outside if the surface is closed. Use the right hand rulewhen a closed line integral on the boundary curve is defined.)
Volume integration: over a volume element dv = dxdydz.
Shinsuke Kawai Vector Calculus
DifferentiationIntegration
Integrations in vector calculusIntegration formulaeMaxwell’s equations
Integration formulae
There are formulae relating integrals over different dimentions.
The divergence theorem:volume integral over ⌃ $ surface integral over ∂⌃Z
⌃
~— ·~Fdv =I
∂⌃~F ·d~A.
Stokes’ theorem:surface integral over ⌃ $ line integral over ∂⌃Z
⌃(~—⇥~
F) ·d~A =I
∂⌃~F ·d~s.
~F =~
F(x ,y ,z) is a vector. ∂⌃ is the boundary of ⌃.
Shinsuke Kawai Vector Calculus
DifferentiationIntegration
Integrations in vector calculusIntegration formulaeMaxwell’s equations
Proof of the divergence theorem
The divergence theorem isZ
⌃
~— ·~Fdv =I
∂⌃~F ·d~A.
Note that the theorem is additive, so it is enough to show it for a smallcube of size dx ⇥dy ⇥dz.
(x0,y0,z0)
(x0,y0+dy,z0)
(x0,y0,z0+dz)
(x0,y0+dy,z0+dz)(x0+dx,y0+dy,z0+dz)
(x0+dx,y0,z0)
(x0+dx,y0,z0+dz)
(x0+dx,y0+dy,z0)
dz
dydx
F(x0,y0,z0)F(x0+dx,y0,z0)→
→
Shinsuke Kawai Vector Calculus
DifferentiationIntegration
Integrations in vector calculusIntegration formulaeMaxwell’s equations
Proof of the divergence theorem – cont.
Using f (x0 +dx) = f (x0)+∂ f
∂x
���x=x0
dx +O(dx
2),
Z
⌃
~— ·~Fdv =Z
⌃
✓∂F
x
∂x
+∂F
y
∂y
+∂F
z
∂z
◆dxdydz
=Z
dydz (Fx
(x0 +dx)�F
x
(x0))
+Z
dzdx (Fy
(y0 +dy)�F
y
(y0))
+Z
dxdy (Fz
(z0 +dz)�F
z
(z0))
=I
∂⌃~F ·d~A.
Shinsuke Kawai Vector Calculus
DifferentiationIntegration
Integrations in vector calculusIntegration formulaeMaxwell’s equations
Proof of Stokes’ theorem
Stokes’ theorem isZ
⌃(~—⇥~
F) ·d~A =I
∂⌃~F ·d~s.
Again, it is enough to show it for a small area of size dx ⇥dy .
dx
dy(x0,y0)
F(x0,y0)→
→
→
→
F(x0+dx,y0+dy)
F(x0+dx,y0)
F(x0,y0+dy)
Shinsuke Kawai Vector Calculus
DifferentiationIntegration
Integrations in vector calculusIntegration formulaeMaxwell’s equations
Proof of Stokes’ theorem – cont.
Z
⌃(~—⇥~
F) ·d~A =Z
⌃
✓∂F
y
∂x
� ∂F
x
∂y
◆dxdy
=Z
dx
∂F
y
∂x
dy �Z
dy
∂F
x
∂y
dx
=Z
(Fy
(x0 +dx ,y ,z)�F
y
(x0,y ,z))dy
�Z
(Fx
(x ,y0 +dy ,z)�F
x
(x ,y0,z))dx
=I
∂⌃~F ·d~s.
Shinsuke Kawai Vector Calculus
DifferentiationIntegration
Integrations in vector calculusIntegration formulaeMaxwell’s equations
Maxwell’s equations – Gauss (electric)
In the differential representation,
~— ·~E =re0
Performing the volume integration and applying the divergencetheorem, Z
⌃
~— ·~Edv =I
∂⌃~E ·d~A =
1e0
Z
⌃rdv .
q =R⌃ rdv is the electric charge contained in the region ⌃. Hence
I
∂⌃~E ·d~A =
q
e0.
Shinsuke Kawai Vector Calculus
DifferentiationIntegration
Integrations in vector calculusIntegration formulaeMaxwell’s equations
Maxwell’s equations – Gauss (magnetic)
In the differential representation,
~— ·~B = 0
Performing the volume integration and applying the divergencetheorem, Z
⌃
~— ·~Bdv =I
∂⌃~B ·d~A = 0.
Shinsuke Kawai Vector Calculus
DifferentiationIntegration
Integrations in vector calculusIntegration formulaeMaxwell’s equations
Maxwell’s equations – Faraday
In the differential representation,
~—⇥~E =�∂~B
∂ t
Performing the surface integration and applying Stokes’ theorem,
Z
⌃(~—⇥~
E) ·d~A =I
∂⌃~E ·d~s =� ∂
∂ t
Z
⌃
~B ·d~A.
�B
=R⌃~B ·d~A is the magnetic flux. Hence,
I
∂⌃~E ·d~s =�∂�
B
∂ t
.
Shinsuke Kawai Vector Calculus
DifferentiationIntegration
Integrations in vector calculusIntegration formulaeMaxwell’s equations
Maxwell’s equations – Ampère-Maxwell
In the differential representation,
~—⇥~B = µ0~j + e0µ0
∂~E∂ t
Performing the surface integration and applying Stokes’ theorem,
Z
⌃(~—⇥~
B) ·d~A =I
∂⌃~B ·d~s = µ0
Z
⌃
~j ·d~A+ e0µ0
∂∂ t
Z
⌃
~E ·d~A.
I =R⌃~j ·d~A is the current and �
E
=R⌃~E ·d~A is the electric flux.
Hence, I
∂⌃~B ·d~s = µ0I + e0µ0
∂�E
∂ t
.
Shinsuke Kawai Vector Calculus
overview of Electromagnetism
Maxwell’s equations1. Gauss (electric)
2. Gauss (magnetic)
3. Faraday
4. Ampère-Maxwell
Coulomb
Biot-Savart
Integral expressions
I~E · d ~A = q/✏0
I~B · d ~A = 0
I~E · d~s = �@�B
@t
I~B · d~s = µ0I + ✏0µ0
@�E
@t
Differential expressions
d ~B =µ0
4⇡
Id~s⇥ ~n
r2
r · ~E = ⇢/✏0
r · ~B = 0
r⇥ ~E = �@ ~B
@t
r⇥ ~B = µ0~j + ✏0µ0
@ ~E
@t
~E =1
4⇡✏0
q
r2~n
~F = q( ~E + ~v ⇥ ~B)Lorentz force
(special case)