Chapter 10 - Divergence And Curl Of Vector Fields

Question 1

Div v

Answer 1

If the components vi, i=1,2,3, of a vector field v=(v1,v2,v3) are continously differentiable functions of the Cartesian coordinates, x,y,z, then the divergence of the vector field is a scalar field defined as
Div v = (dv1/dx) + (dv2/dy) + (dv3/dz)

Question 2

Curl v

Answer 2

Curl v =
(dv3/dy - dv2/dz, dv1/dz - dv3/dx, dv2/dx - dv1/dy)

To remember: determinant of
I j k
D/dx d/dy d/dz
V1 v2 v3

Question 3

Div (lamda v + lamda’ v’)

Curl (lamda v + lamda’ v’)

Answer 3

= lamdadiv(v) + lamda’div(v’)

= lamdacurl (v) + lamda’curl (v’)

Question 4

When does curl v = 0?

Answer 4

When a continuous differentiable vector field v can be written as the gradient of a scalar field phi,
i.e. phi is a potential function for v

Question 5

Irrotational on R

Answer 5

A vector field v for which curl v = 0 in some region R (a subset of R^3)

Question 6

Scalar potential for the field v

Answer 6

If v is an irrotational vector field on a singly-connected region R (subset of R^3), then there exists a scalar field phi such that v=grad of phi on R. This phi is the scalar potential field.

Question 7

When does div (curl v) = 0?

Answer 7

If the vector field v has continuous second derivatives

Question 8

Solenoidal vector field

Answer 8

Div v = 0

Question 9

Vector potential for v

Answer 9

If v is a solenoidal vector field on R, then there exists a vector field A such that v = curlA on R.
A is the vector potential for v.
If A is a vector potential for v, so is A+grad (phi), for any scalar field phi

Question 10

Div v and curl v in terms of grad

Answer 10

Div v = grad•v

Curl v = grad cross product v

Question 11

Laplacian operator: (grad)^2

Answer 11

“Del squared” = (d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)

Question 12

Del squared*phi

Del squared*v

Answer 12

= div (grad phi)

= grad (div v) - curl (curl v)