za vector calculus - university of...
TRANSCRIPT
Adapted from notes by Prof. Stuart A. Long
Notes 3 Review of
Vector Calculus
1
ECE 3317Applied Electromagnetic Waves
Prof. David R. JacksonFall 2018
ˆ zzA
ˆ yyA
ˆ xxA V x y z∆ = ∆ ∆ ∆
y
z
x
Overview
2
Here we present a brief overview of vector calculus. A much more thorough discussion of vector calculus may be found in the class notes for ECE 3318:
http://courses.egr.uh.edu/ECE/ECE3318
Notes 13: DivergenceNotes 17: CurlNotes 19: Gradient and Laplacian
Please also see the textbooks and the following supplementary books(on reserve in the Library):
H. M. Schey, Div, Grad, Curl, and All That: an Informal Text on Vector Calculus, 2nd Ed.,W. W. Norton and Company, 1992.
M. R. Spiegel, Schaum’s Outline on Vector Analysis, McGraw-Hill, 1959.
Gradient
“Del” Operator
ˆ ˆ ˆx y zx y zφ φ φφ ∂ ∂ ∂
∇ = + +∂ ∂ ∂
( )2 2 2
22 2 2x y z
φ φ φ ∂ ∂ ∂
= ∇ ⋅∇ = + + ∂ ∂ ∂ ∇Laplacian
ˆ ˆ ˆ x y zx y z∂ ∂ ∂
≡ + +∂ ∂ ∂
∇
(Scalar)
(Vector)
This is an “operator”.
3
Divergence yx zAA AA
x y z∂∂ ∂
∇⋅ = + +∂ ∂ ∂
ˆ ˆ ˆy yz x z xA AA A A AA x y zy z z x x y
∂ ∂ ∂ ∂ ∂ ∂ ∇× = − + − + − ∂ ∂ ∂ ∂ ∂ ∂
Vector A: ˆ ˆ ˆx y zA A x A y A z= + +
ˆ ˆ ˆ x y zx y z∂ ∂ ∂
= + +∂ ∂ ∂
∇
“Del” Operator (cont.)
Curl
(Vector)
(Scalar)
Note:Results for cylindrical and spherical coordinates are given
in the back of your books.4
ˆ ˆ ˆx y zA A x A y A z= + +
ˆ ˆ ˆ x y zx y z∂ ∂ ∂
= + +∂ ∂ ∂
∇
“Del” Operator (cont.)
5
( )ˆ ˆ ˆ ˆˆ ˆ
ˆ ˆ ˆ
ˆ ˆ ˆ
x y z
x y z
y yx xz z
A x y z x A y A z Ax y z
x y z
x y zA A A
A AA AA Ax y zy z x z x y
∂ ∂ ∂∇× = + + × + + ∂ ∂ ∂
∂ ∂ ∂=∂ ∂ ∂
∂ ∂ ∂ ∂∂ ∂ = − − − + − ∂ ∂ ∂ ∂ ∂ ∂
( )ˆ ˆ ˆ ˆˆ ˆ yx zx y z
AA AA x y z x A y A z Ax y z x y z
∂ ∂ ∂∂ ∂ ∂∇⋅ = + + ⋅ + + = + + ∂ ∂ ∂ ∂ ∂ ∂
A few more details about calculating the divergence and the curl:
( )
( )
0
0
A
φ
∇ ⋅ ∇× =
∇× ∇ =
Vector Identities
Two fundamental “zero” identities:
6
Vector Identities (cont.)
( ) ( ) ( )A B B A A B∇⋅ × = ⋅ ∇× − ⋅ ∇×
Another useful identity:
This will be useful in the derivation of the Poynting theorem.
7
( ) ( )2 A A A∇ ≡∇ ∇⋅ ∇× ∇×−
Vector Laplacian
The vector Laplacian of a vector function is a vector function.
The vector Laplacian is very useful for deriving the vector Helmholtz equation (the fundamental differential equation that the electric and magnetic fields obey).
8
( ) ( ) ( )2 2 2 2ˆ ˆ ˆx y zA= x A y A z A∇ ∇ + ∇ + ∇
Vector Laplacian (cont.)
In rectangular coordinates, the vector Laplacian has a very nice property:
This identity is a key property that will help us reduce the vector Helmholtz equation to the scalar Helmholtz equation, which the components of the fields satisfy.
9
ˆ ˆ ˆ= x y zx y zφ φ φφ
∂ ∂ ∂ ∇ + + ∂ ∂ ∂
Gradient
( ) ( ) ( )ˆ ˆ ˆ ˆˆ ˆ
d = dx dy dzx y z
x y z x dx y dy z dz drx y z
φ φ φφ
φ φ φ φ
∂ ∂ ∂+ +
∂ ∂ ∂
∂ ∂ ∂ = + + ⋅ + + = ∇ ⋅ ∂ ∂ ∂
d dr dr=d d dφ φ φ∇ ⋅ = ∇ ⋅
( )d =dφ φ∇ ⋅
The gradient vector tells us the direction of maximum change in a function. 10
(from calculus)
φ∇
d
dr
r r dr+
Gradient (cont.)
11
ˆ ˆ ˆx y zx y z
∂Φ ∂Φ ∂Φ∇Φ = + +
∂ ∂ ∂
1 1ˆ ˆˆsin
rr r r
θ φθ θ φ
∂Φ ∂Φ ∂Φ∇Φ = + +
∂ ∂ ∂
1ˆˆ zz
ρ φρ ρ φ
∂Φ ∂Φ ∂Φ∇Φ = + +
∂ ∂ ∂
Rectangular
Cylindrical
Spherical
0
1 ˆLimV
S
A A n dSV∆ →
∇ ⋅ = ⋅∆ ∫
Divergence
The divergence measures the rate at which the “flux” of the
vector function emanates from a region of space.
12
Please see the books or the ECE 3318 class notes for a derivation of this property.
ˆ zzA
ˆ yyA
ˆ xxA V x y z∆ = ∆ ∆ ∆
y
z
x
Divergence > 0: “source of flux”Divergence < 0: “sink of flux”
ˆ = n outward normal
Divergence (cont.)
13
Many small pipes injecting water
Tub of water
Water flow
A(x,y,z) = velocity vector of water inside tub
0A∇⋅ >
Divergence (cont.)
14
Rectangular:
Cylindrical:
Spherical:
yx zAA AA
x y z∂∂ ∂
∇ ⋅ = + +∂ ∂ ∂
( )1 1 zA AA Az
φρρ
ρ ρ ρ φ∂ ∂∂
∇ ⋅ = + +∂ ∂ ∂
( ) ( )22
1 1 1sinsin sinr
AA r A A
r r r rφ
θ θθ θ θ φ
∂∂ ∂∇ ⋅ = + +
∂ ∂ ∂
( ), 0
1ˆ Limy z
C
A x A dry z∆ ∆ →
∇× ⋅ = ⋅∆ ∆ ∫
Curl
15
A component of the curl tells us the rotation of the vector function about that axis.
x
, 0y z∆ ∆ →
"right-hand rule for C"
z
y
z∆
y∆
C
xS y z∆ =∆ ∆
Please see the books or the ECE 3318 class notes for a derivation of this property.
RiverPaddle wheel
( ) ˆ 0A x <∇× ⋅
y
z
C
( )A A dr= ⋅velocity vector gives force on vanes
( )
( )
( )
0
0
0
1ˆ Lim
1ˆ Lim
1ˆ Lim
xx
xy
xz
Sx C
Sy C
Sz C
A x A drS
A y A drS
A z A drS
∆ →
∆ →
∆ →
∇× ⋅ ≡ ⋅∆
∇× ⋅ ≡ ⋅∆
∇× ⋅ ≡ ⋅∆
∫
∫
∫
Curl (cont.)
16
Curl is calculated here
x
y
z
xC
yC
zC
xS∆
yS∆
zS∆
Note: The paths are all centered at the point of interest
(a separation between them is shown in the exploded view for clarity).
Note:The paths are defined according to the
“right-hand rule.”
“Exploded view”
Curl (cont.)
17
( )1 1ˆˆ ˆz zAA A AA AA z
z zφφ ρ ρρ
ρ φρ φ ρ ρ ρ φ
∂∂ ∂ ∂ ∂ ∂ ∇× = − + − + − ∂ ∂ ∂ ∂ ∂ ∂
( )
( ) ( )sin1 1 1 1ˆˆsin sin
r rA rA rAA A AA r
r r r r rφ φ θθ
θθ φ
θ θ φ θ φ θ
∂ ∂ ∂ ∂ ∂ ∂∇× = − + − + − ∂ ∂ ∂ ∂ ∂ ∂
ˆ ˆ ˆy yx xz zA AA AA AA x y z
y z z x x y∂ ∂ ∂ ∂∂ ∂ ∇× = − + − + − ∂ ∂ ∂ ∂ ∂ ∂
Rectangular
Cylindrical
Spherical
Divergence Theorem
18
ˆV S
A dV A n dS∇⋅ = ⋅∫ ∫
A = arbitrary vector function
n = outward normal
nV
S
Stokes’s Theorem
19
The unit normal is chosen from a “right-hand rule”according to the direction along C.
(An outward normal corresponds to a counter clockwise path.)
( ) ˆS C
A n dS A dr∇× ⋅ = ⋅∫ ∫
C (closed)
S (open)n