Gradient and Divergence

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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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