Engineering Math: Lecture Notes on Gradient, Divergence, and Surface Normals, Lecture notes of Engineering Mathematics

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MAT 247 Engineering Mathematics

Hakkı Ula¸s Unal¨ Dept. of Electrical-Electronics Eng. Anadolu University, Turkey

October 3, 2018

Today

1 Gradient Operator:Review

(^2) Properties of gradient

(^3) Divergence of a Vector field

Some Review on Gradient Operator:

Difference between scalar and vector fields:

-14.5562 -42.

-14.

-14.

12.^

(^00 1 2 3 )

1

2

B A^ A

gradf = ∂f ∂x

i + ∂f ∂y

j,

∂f ∂x ∂f : Rate of change along^ x^ direction for fixed^ y: ∂y : Rate of change along^ y^ direction for fixed^ x: What about moving from A to B, say A is fixed and B varying, however, it is parallel to unit vector?

Example

Find the directional derivative of f (x, y, z) = 2x^2 + 3y^2 + z^2 at the point A in the direction of b = i − 2 k.

Direction of maximum increase

Let f (P ) = f (x, y, z) be a scalar function having continuos partial derivatives in some domain in the space

∇f = ∂f ∂x

i + ∂f ∂y

j + ∂f ∂z

k,

Direction of maximum increase

Let f (P ) = f (x, y, z) be a scalar function having continuos partial derivatives in some domain in the space

∇f = ∂f ∂x

i + ∂f ∂y

j + ∂f ∂z

k,

Then, the directional derivative

Dbf = (^) ds df (x(s), y(s), z(s)) = b·∇f = |b||∇f | cos θ,

where θ is a positive angle btw ∇f and b. Then, Dbf = |gradf |

Gradients as a surface normal vector

Lemma

Suppose f (x, y, z) is a scalar field and let f (x, y, z) = c, where c is constant, and let P (x, y, z) be a particular point on the surface. Then, gradf , if it is non-zero at P , is a normal vector to the level surface f = c at P.

Gradients as a surface normal vector

Lemma

Suppose f (x, y, z) is a scalar field and let f (x, y, z) = c, where c is constant, and let P (x, y, z) be a particular point on the surface. Then, gradf , if it is non-zero at P , is a normal vector to the level surface f = c at P.

Proof

Suppose that r(t) = [x(t), y(t), z(t)] represents a curve C passing through P and lying on the level surface f = c. Then, r(t) must satisfy

f (x(t), y(t), z(t)) = c.

Example

Find a unit normal vector for the curve described as

log(x^2 + y^2 ) − 4 = 0,

at P = (2, 0).

Divergence of a Vector field

Let v(x, y, z) be a differentiable vector function, where x, y, z are Coordinates and let v 1 , v 2 , v 3 are components of v.

div v = ∂v^1 ∂x

• ∂v^2 ∂y

• ∂v^3 ∂z

is called the divergence of v.

Curl of a Vector field

Let v(x, y, z) = [v 1 , v 2 , v 3 ] = v 1 i + v 2 j + v 3 k be a differentiable vector function, where x, y, z are Coordinates, and v 1 , v 2 , v 3 are components of v. Then the curl of the vector function v is defined as

curlv = ∇ × v =

i j k ∂ ∂x

∂ ∂y

∂ ∂z v 1 v 2 v 3

∂v 3 ∂y

− ∂v^2 ∂z

i +

∂v 1 ∂z

− ∂v^3 ∂x

j +

∂v 2 ∂x

− ∂v^1 ∂y

k.

Example

Find curlv and curl curlv, where v = (x + z, x + y, y + z).