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Gradient and Divergence

Dr Aiden Price

EGB241

Week 9

1/13

Gradient and

Divergence

EGB241

Gradient of a

Scalar Field

Divergence

Next Week

2/13

Gradient of a Scalar Field

• A gradient is another term for derivative.

• Consider a univariable function (one input), f(x).

• f(x) has one variable and one derivative; dx df .

• Consider a multivariable function (this time with three

inputs), f(x, y, z).

• f(x, y, z) has three variables and so has three partial

derivatives; ∂f ∂x, ∂f ∂y and ∂f ∂z .

Gradient and

Divergence

EGB241

Gradient of a

Scalar Field

Divergence

Next Week

3/13

Gradients II

• The first mathematical definition is the Del operator, ∇.

In Cartesian coordinates:

∇ = ∂

∂xxˆ + ∂y ∂ yˆ + ∂z ∂ z. ˆ

• When applied to some quantity, T(x, y, z), the gradient

of T is written:

∇T = ∂T

∂x xˆ + ∂T ∂y yˆ + ∂T ∂z z. ˆ

• This vector points in the direction of greatest increase in

T (positive → go forwards, negative → go back).

Gradient and

Divergence

EGB241

Gradient of a

Scalar Field

Divergence

Next Week

4/13

Directional Derivatives

• Consider the derivative along direction dl = dlaˆl, where

aˆl = l

|l|. Then

dT

dl = ∇T · aˆl.

• The difference in T between two points P1 and P2 along

dl can instead be expressed as

T2 – T1 = ZPP12 ∇T · dl = ZPP12 ∇T · aˆldl.

Gradient and

Divergence

EGB241

Gradient of a

Scalar Field

Divergence

Next Week

5/13

Gradients – Example I

• Find the directional derivative of T = x2 + y2z along

direction l = 2ˆ x + 3ˆ y – 2ˆ z and evaluate it at (1, -1, 2).

Gradient and

Divergence

EGB241

Gradient of a

Scalar Field

Divergence

Next Week

6/13

Gradients: Coordinate Systems

• Cartesian:

• ∇ = ∂

∂xxˆ + ∂y ∂ yˆ + ∂z ∂ z. ˆ

• ∇T = ∂T

∂x xˆ + ∂T ∂y yˆ + ∂T ∂z z. ˆ

• Cylindrical:

• ∇ = ∂

∂r rˆ + 1r ∂φ ∂ φˆ + ∂z ∂ z. ˆ

• ∇T = ∂T

∂r rˆ + 1r ∂T ∂φ φˆ + ∂T ∂z z. ˆ

• Spherical:

• ∇ = ∂

∂R

Rˆ + 1

R

∂

∂θ

θˆ+

1

Rsinθ

∂

∂φ

ˆ φ.

• ∇T = ∂T

∂R

Rˆ + 1

R

∂T

∂θ

θˆ+ 1

Rsinθ

∂T

∂φ

ˆ φ.

Gradient and

Divergence

EGB241

Gradient of a

Scalar Field

Divergence

Next Week

7/13

Gradients – Example II

• Find the gradient of the following scalar functions and

evaluate at the given point:

(a) V1 = 24V0cos(πy 3 )sin(2πz 3 ) at (3, 2, 1) in Cart. coords.

(b) V2 = V0e-2rsin(3φ) at (1, π2 , 3) in cylindrical coords.

(c) V3 = V0(Ra )cos(2θ) at (2a, 0, π) in spherical coords.

Gradient and

Divergence

EGB241

Gradient of a

Scalar Field

Divergence

Next Week

8/13

Divergence and Flux I

• Divergence of a vector, E, at a point is the net outward

flux per unit volume over a closed surface.

• Flux is a vector describing the magnitude and flow of

some substance or property (such as electric field) through

a surface.

Gradient and

Divergence

EGB241

Gradient of a

Scalar Field

Divergence

Next Week

9/13

Divergence and Flux II

• Divergence of E:

∇ · E = ∂Ex

∂x +

∂E

y

∂y +

∂Ez

∂z .

• Total flux:

IS E · ds.

• Divergence theorem:

ZV ∇ · EdV = IS E · ds.

Gradient and

Divergence

EGB241

Gradient of a

Scalar Field

Divergence

Next Week

10/13

Divergence: Coordinate Systems

• Cartesian:

∇ · E = ∂Ex

∂x +

∂E

y

∂y +

∂Ez

∂z .

• Cylindrical:

∇ · E = 1

r

∂

∂r(rEr) + 1r ∂E ∂φφ + ∂E ∂zz .

• Spherical:

∇·E = 1

R2

∂

∂R(R2ER)+Rsinθ 1 ∂θ ∂ (sinθEθ)+Rsinθ 1 ∂E ∂φφ .

Gradient and

Divergence

EGB241

Gradient of a

Scalar Field

Divergence

Next Week

11/13

Divergence – Example I

• Determine the divergence of each of the following vector

fields:

(a) E = 3x2xˆ + 2zyˆ + x2zzˆ at (2, -2, 0).

(b) E = a3Rcosθ 2 Rˆ – a3Rsinθ 2 θˆ at (a2, 0, π).

Gradient and

Divergence

EGB241

Gradient of a

Scalar Field

Divergence

Next Week

12/13

Divergence – Example II

• Given E = kRRˆ, calculate the flux of E through a

spherical surface of radius a, centred at the origin.

Gradient and

Divergence

EGB241

Gradient of a

Scalar Field

Divergence

Next Week

13/13

Next Week

• Circulation (line integrals).

• Curl.

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