2. Vector Differentiation

Question 1

Definition of the Partial Derivative

Answer 1

∂f/∂x = (ε->0)lim f(x+ε, y, z) - f(x,y,z) / ε

-it is the derivative with respect to one variable whilst treating all other variables as constants

Question 2

Exact Derivative

Definition

Answer 2

df = ∂f/∂x dx + ∂f/∂y dy + ∂f/∂z dz

Question 3

The Chain Rule

x, y, z as functions of s

Answer 3

-let x, y, z be continuously differentiable functions of s, then:
df/ds = ∂f/∂x dx/ds + ∂f/∂y dy/ds + ∂f/∂z dz/ds

Question 4

The Chain Rule

x, y, z as functions of u, v, w

Answer 4

∂f/∂u = ∂f/∂x ∂x/∂u + ∂f/∂y ∂y/∂u + ∂f/∂z ∂z/∂u

∂f/∂v = ∂f/∂x ∂x/∂v + ∂f/∂y ∂y/∂v + ∂f/∂z ∂z/∂v

∂f/∂w = ∂f/∂x ∂x/∂w + ∂f/∂y ∂y/∂w + ∂f/∂z ∂z/∂w

Question 5

Simple Differentiation of Vectors

Answer 5

-consider a vector |F that is a function of a single variable t:
|F(t) = f1(t)e1 + f2(t)e2 + f3(t)e3
-for fixed cartesian axes {e1/e2/e3} we have:
d|F/dt = df1/dt e1 + df2/dt e2 + df3/dt e3

Question 6

Simple Differentiation of Vectors - Notes

Answer 6

i) d|F/dt is a vector
ii) the equation for simple differentiation of vectors only applies to constant basis vectors (i.e. fixed axes), if e1, e2, e3 also change with t then they also have to be differentiated
iii) obvious generalisation for higher derivatives

Question 7

Tangent Vectors

Answer 7

• the curve with position vector |r(s) where s is a parameter increasing along the curve, has tangent vector d|r.ds
• hence the unit tangent vector is (d|r/ds) / | d|r/ds |

Question 8

Field Definition

Answer 8

• a scalar or vector that is a function of position is called a field
  e. g. temperature is a scalar field

Question 9

Contours

Answer 9

• in 2D, fields can be represented by contours
• contours are curves of equal value of the field
• in 3D contours are curved surfaces defined by T(|r)=T0
• this is a surface that can be written in the form f(x,y,z)=constant

Question 10

Vector Fields

Answer 10

• an example of a vector field is wind speed which can be written |u(|r)
• vector fields can be represented by arrows with size and direction representing ||u| and |u^

Question 11

Gradient of a Scalar Field

Answer 11

-if f(x,y,z) is a continuously differentiable scalar field, then its “gradient”, written grad f = ∇f is defined as:
∇f = ∂f/∂x e1 + ∂f/∂y e2 + ∂f/∂z e3
= (∂f/∂x, ∂f/∂y, ∂f,∂z)

Question 12

Notes on Grad

Answer 12

i) grad f is a vector

ii) ∇(f + g) = ∇f + ∇g

Question 13

Geometric Interpretation of Grad

Formula

Answer 13

-consider a scalar field f(|r)
-the change in f on moving from |r to (|r+d|r)is found from the exact derivative:
df = ∂f/∂x dx + ∂f/∂y dy + ∂f//∂z dz -> a scalar product
df = (∂f/∂x, ∂f/∂y, ∂f/∂z) . (dx, dy, dz)
df = (∇f) . d|r

Question 14

Geometric Interpretation of Grad

Case 1: d|r parallel to contour

Answer 14

• consider a contour of f passing through a point at |r, let d|r lie in the tangent plane of the contour
• then df=0 as you move along the contour as f is constant along the contour
• > (∇f) . d|r = 0
• hence d|r is perpendicular to ∇f
• so ∇f is normal to all vectors in the tangent plane of a contour -> ∇f is normal to any surface of constant f

Question 15

Geometric Interpretation of Grad

Case 2: d|r perpendicular to the contour

Answer 15

-let d|r be perpendicular to the contour of f
-let d|r = n^ds where ds is the magnitude of d|r and n^ is a unit normal to the contour
-> df = (∇f) . n^ ds
-but ∇f is also normal to the contour so (∇f) . n^ = |∇f||n|
-> df = |∇f| ds
|∇f| = df/ds
-so ∇f is the rate of change of f in the direction in which f is changing most rapidly
-the magnitude and direction of ∇f are coordinate invariant

Question 16

Directional Derivative

Equation

Answer 16

df/ds = u^ . ∇f

Question 17

Directional Derivative

Derivation

Answer 17

-the directional derivative of a field, f, in the direction of unit vector u^, can be found by substituting into the grad formula:
df = (∇f) . d|r
-in this case we take 
d|r = u^ ds
-where ds is the magnitude of dr
df = (∇f) . u^ ds
-the dot product commutes
df/ds = u^ . ∇f

Question 18

Directional Derivative

Notes

Answer 18

-the directional derivative of a scalar field is a scalar field

Question 19

The Del Operator

Answer 19

-∇ is an operator that operates on functions to produce other functions
∇ = e1 ∂/∂x + e2 ∂/∂y + e3 ∂/∂z
-this is a vector differential operator called ‘del’, and its symbol is called nabla
-the del operator is coordinate invariant

Question 20

The Del Operator and Grad

Answer 20

-we can interpret grad f as the result of operating with ∇ on s scalar field f ro produce a vector field

Question 21

Divergence

Definition

Answer 21

-the divergence of a field
|F(|r) = (F1(|r), F2(|r), F3(|r)) is defined:
div F = ∂F1/∂x + ∂F2/∂y + ∂F3/∂z

Question 22

The Del Operator and Divergence

Answer 22

div F = ∇ . F

Question 23

Divergence

Derivation

Answer 23

div F = ∇ . F =
(∂/∂x, ∂/∂y, ∂/∂z) . (F1, F2, F3)
=∂F1/∂x + ∂F2/∂y + ∂F3/∂z

Question 24

Divergence

Notes

Answer 24

• the divergence of a vector field is a scalar field
• ∇ is also a linear operator so ∇.(F+G)=∇.F + ∇.G
• a field |F for which ∇.F=0 is called ‘incompressible’ OR ‘solenoidal’

Question 25

Physical Interpretation of Divergence

Answer 25

-if |F is a field (velocity of a fluid) then div |F is the net flow OUT of any given location

Question 26

Curl of a Vector Field

Definition

Answer 26

curl |F = ∇ x |F

Question 27

Curl of a Vector Field

Notes

Answer 27

-curl |F is a vector field

Question 28

Physical Interpretation of Curl

Answer 28

• curl measures the rotation or twisting of a vector field
• a field that rotates anticlockwise in the x-y plane has a curl in the positive z direction (right hand rule)
• a field for which curl |F = |0 every where is called irrotational

Question 29

More Uses of Del

Answer 29

(F . ∇)G where (F.∇) acts on scalar field G
Fx∇
∇ . ∇

Question 30

The Laplacian Operator

Definition

Answer 30

-if we form ∇.∇ we get a second-order, scalar differential operator

∇.∇ = ∇² = ∂²/∂x² + ∂²/∂y² + ∂²/∂z²

Question 31

The Laplacian Operator

Notes

Answer 31

• the laplacian can act on both scalars and vectors
• when acting on a scalar field the laplacian gives a scalar
• when acting on a vector field, the laplacian gives a vector

Question 32

Applications of Vector Calculus

Heat Flow

Answer 32

a) Fourier’s Law - heat flows from hot to cold at a rate proportional to temperature gradient, gradT points in the direction of steepest temperature gradient so heat flow is in the opposite direction to this and proportional to magnitude
Heat Flux = |J = -k ∇T
b) the rate of change of temperature at a point is proportional to the rate at which heat flux converges, temperature wont change is the net flow of heat in equals net flow of heat out i.e. if divJ=0
eCp ∂T/∂t = - ∇.J
-combining (a) and (b) gives The Diffusion Equation
∂T/∂t = k/eCp * ∇²T

Question 33

Applications of Vector Calculus

Fluid Mechanics

Answer 33

-in an incompressible fluid, the velocity |u and pressure p are related by:
∇ . |u = 0
-since for an incompressible fluid the flow into any point is equal to the flow out of any point

Question 34

Applications of Vector Calculus

Electromagnetism

Answer 34

-electric and magnetic fields, |E and |B, are governed by Maxwell's equations:
∇ . |E = Q/ε0
∇ . |B = 0
∇x|E = -∂B/∂t
∇x|B = μ0f + μ0ε0 *∂|E/∂t