Lectures 10, 11 and 12

Question 1

What is the change in volume element dV when an area element is revolved by dθ?

Answer 1

dV = dAydθ

Question 2

What is the integral equation for the volume of revolution?

Answer 2

V = π * integral from x1 to x2 of f(x)^2 dx

Question 3

What is Pappus’ first theorem?

Answer 3

V = 2πAy(centroid)

Question 4

What is the divergence of a vector and what is the divergence symbol?

Answer 4

∇.v, where v is a vector. ∇ = d/dx i + d/dy j + d/dz k

Question 5

How do we interpret the divergence of say an electric field when it is equal to zero?

Answer 5

Net flux of field lines through a volume - same number of lines enter as leave so is zero.

Question 6

What is the divergence of a magnetic field in all space?

Answer 6

Zero.

Question 7

What is the curl of a vector and how is it calculated?

Answer 7

∇ X v, where v is a vector. Equals the determinant with the three vector components along the bottom, i,j,k along the top and d/dx, d/dy and d/dz along the middle.

Question 8

What are two cases with a raft in water where the curl is and is not equal to zero respectively? What happens when the curl is not equal to zero?

Answer 8

Equals zero when v = x i

Not equal to zero when v = x j - in this case the raft will tend to turn to the left.

Question 9

What is the curl equal to for a whirlpool? What happens to a paddle wheel if you put it in the whirlpool?

Answer 9

Zero. A paddle wheel will circulate around the centre but will not spin.

Question 10

What is grad squared also known as?

Answer 10

The Laplacian.

Question 11

What is Laplace’s equation?

Answer 11

∇^2 * Ф = 0

Question 12

What is Poisson’s equation?

Answer 12

∇^2 * Ф = k

Question 13

What is curl grad phi? How is it calculated?

Answer 13

∇ X ∇*Ф. Calculated with the determinant but along the bottom is the differentiated vector components rather than normal.

Question 14

What is curl grad phi always equal to?

Answer 14

Zero.

Question 15

What is div curl A and how is it calculated?

Answer 15

∇*(∇ X A). Calculated by doing the curl part first then taking divergence of each component.

Question 16

What is div curl A equal to?

Answer 16

Zero.

Question 17

What is the gradient of a divergence and how is it calculated?

Answer 17

∇*(∇ * A). Calculated by taking the divergence of a first and then multiplying it all out.

Question 18

What is the gradient of a divergence equal to?

Answer 18

A vector.

Question 19

What is curl curl A?

Answer 19

∇ X (∇ X A)

Question 20

What is another way of writing curl curl A?

Answer 20

∇^2 A = ∇*(∇ * A) - ∇ X (∇ X A)

Question 21

How can we calculated ∇*(fg), where f and g are scalars?

Answer 21

Use the product rule:∇(fg) = f(∇g) + g(∇*f)

Question 22

How can we calculate ∇*(fg), where f is a scalar and g is a vector?

Answer 22

∇(fg) = g * ∇f + f(∇*g)