∇ Acting on Vector Fields

Question 1

What does ∇.v stand for?

Answer 1

The divergence of v

Question 2

What is the symbol for the divergence of v?

Answer 2

∇.v

Question 3

What is the equation for the divergence of v, ∇.v?

Answer 3

Question 4

If ∇.v a scalar or vector field? And why?

Answer 4

Scalar field - it maps each point (x, y, z) to a number not a vector

Question 5

What are two properties of divergence?

Answer 5

Question 6

Prove the following property of divergence.

Answer 6

Question 7

Prove the following property of divergence.

Answer 7

Question 8

What is the vector cross product of two vectors A and B?

Answer 8

Question 9

What does ∇ x v stand for?

Answer 9

The curl of v

Question 10

What is the symbol for the curl of v?

Answer 10

∇ x v

Question 11

What is the equation for the curl of v, ∇ x v?

Answer 11

Question 12

What is the curl a measure of?

Answer 12

How much v is curling around

Question 13

Why can you calculate the divergence of a curl?

Answer 13

Because of curl is a vector field

Question 14

What are two properties of curl?

Answer 14

Question 15

Prove the following property of the curl.

Answer 15

Follows from linearity of derivatives - need to do

Question 16

Prove the following propety of the curl.

Answer 16

Question 17

What does the following stand for? And what is another way to write it?

Answer 17

Question 18

What is the equation for the div grad f?

Answer 18

Question 19

What is another name for ∇²f other than div grad f?

Answer 19

Laplacian of f

Question 20

For n=3, in general what does ∇ x _∇_f equal?

Answer 20

Question 21

For n=3, in general what does ∇.(∇ x v) equal?

Answer 21