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