6) Gradient If A Scalar Field

Question 1

Scalar field as a map

Answer 1

Scalar field φ is a C^1 map,
F:S_1 -> S_2 where
S_1 c R^3 and S_2 c R

Question 2

C^k function

Answer 2

C^k function

Function has k derivatives that are all continuous functions

Question 3

The derivative matrix and gradient grad

Answer 3

D(F)(x,y,z) = ( ∂F ∂F ∂F )
– – –
∂x ∂y ∂z
Hence we can use the chain rule to be calculated with respect to ant curvilinear coords

= (F_x, F_y, F_z)

grad F = ( ∂F ∂F ∂F )
– – –
∂x ∂y ∂z

Hence is a linear map
Has linear property
Grad(aφ_1 +bφ_2) = a grad(φ_1) + b grad(φ_2)

Question 4

Gradient at point

Answer 4

At point p gradient is

(grad φ)_p. Or. (Nabla φ)_p

Question 5

Directional derivative

Answer 5

Directional derivativea measure the rate of variation of φ in the direction of the vector 
^
u
-
∇ _u φ = Lim     ( φ (P') - φ(P) )
               h->0   (-------------)
                                       h

The gradient of the scalar field allows us to determine the directional derivative 
               ^
∇ _u φ = u • ( grad φ )_p
               -
Rate of change in direction u

E.g. u = i, in the direction of the x-axis

DOT PRODUCT
so for theta between vector grad φ and u
∇ _u φ = | gradφ|cos(theta)

Unit vector u
As direction u varies so does theta and maximum value at p occurs for theta=0

Question 6

Maximum rate of increase in a direction

Answer 6

Dot product of grad and unit vector u

Max rate of decrease when cos theta = -1
(-grad φ)_p

Question 7

Stationary point?

Answer 7

A stationary point is a point at which (grad φ )_p =vector(0)

Since ∇ u φ =0
For all directions ^u

Hence this implies
(grad φ)p =0
And thus ^u • (grad φ)p = 0 for all ^u

Question 8

Level set

Answer 8

The level set of a map F:R^n -> R correspond to some constant c∈R is the set
L_c(F) = { (x_1, x_2, …, x_n) | F(x_1,…,x_n)= c}
Scalar field and is a level surface ,ISO surface or contour line

E.g. Set field equal to constant

Question 9

Theorem on level surface and grad

Answer 9

Theorem: Let P be any point in the level surface L_c(φ)

L_c( φ) = { (x,y,z) | φ(x,y,z) =0 }

Such that (grad φ)_p not equal to vector(0)
Then
(grad φ)_p = | (grad φ)p | ^n

Where ^n_ is a unit vector normal to the level surface at p

∇ _u φ

Question 10

grad φ and tangent of any curve c in L_c( φ)

Answer 10

Tangent of any curve c and grad phi are ORTHOGONAL

for any curve c in L_c( φ) passing through P

Hence (grad φ) • ^T_ = 0
Where ^T_ = d(vector (r)) /dt

grad φ lies along normal vector to L_c( φ)

Since (grad φ)_p not equal to 0 vector

E.g. Find a normal at P to ellipsis

x^2 y^2. z^2.
— +. —- + —- = k^2 for k in R
a^2. b^2. c^2

By theorem the (grad φ)_p is a normal vector to the surface, the level set of k^2

φ(x,y,z) =. x^2 y^2. z^2.
— +. —- + —-
a^2. b^2. c^2

So non unit normal is (grad φ)_p