Lectures 7, 8 and 9

Question 1

What is an important equation for velocity in cylindrical polars?

Answer 1

v = dρ/dt * eρ + ρ*dϕ/dt * eϕ + dz/dt * ez

Question 2

What is x, y and z equal to in spherical polar coordinates?

Answer 2

x = rsinθcosϕ
y = rsinθsinϕ
z = rcosθ

Question 3

In spherical polars, what are θ and ϕ measured between? Which plane is ϕ measured in? What is θ a measurement of?

Answer 3

0 <= θ <= π
0 <= ϕ <= 2π

ϕ is measured in the x-y plane.
θ is the angle from the z-axis we are looking at

Question 4

Define the changes in surface and volume, ds and dv for spherical polars.

Answer 4

ds = r^2 * sinθ dϕdθ
dv = r^2 * sinθ drdϕdθ

Question 5

What are the three basis vectors for spherical polars?

Answer 5

er, eϕ and eθ

Question 6

What is the equation for the basis vector er?

Answer 6

er = cosϕsinθ i + sinϕsinθ j + cosθ k

Question 7

What is the equation for the basis vector eϕ?

Answer 7

eϕ = - i + cosϕ j

Question 8

What is the equation for the basis vector eθ?

Answer 8

eθ = cosθcosϕ i + cosθsinϕ j - sinθ k

Question 9

For a 1D integral, what is the change in step ds equal to?

Answer 9

ds^2 = dx^2 + dy^2

Question 10

What is an alternative version to write ds?

Answer 10

ds = dx*sqrt(1+(dy/dx)^2), or with dy out the front and dx/dy in the brackets

Question 11

What is a Jacobian?

Answer 11

The transformation from denoting coordinates as one thing to another.

Question 12

Give an example of a Jacobian.

Answer 12

The Jacobian of x,y,z with respect to u,v,w is: J = d(x,y,z)/d(u,v,w)

Question 13

What can we write out change in volume element dv as including the Jacobian?

Answer 13

The magnitude of the Jacobian multiplied by the changed letters: dv = dxdydz = |J|dudvdw

Question 14

How would you calculate the Jacobian?

Answer 14

It is the determinant including the variables involved. e.g. if J = d(x,y)/d(ρ,ϕ), then it is the determinant of these components.

Question 15

Why is the Jacobian useful?

Answer 15

It is an easier way of working out the coordinate system transformations for dA and dV etc, by using the determinant and substituting in the corresponding conversion equations.

Question 16

What are hyperbolic coordinates?

Answer 16

Where you convert the coordinates by using the two axis, for example x,y, and using xy = r, and x/y = s, giving a simple square.

Question 17

In a surface of revolution, what is the change in area dA equal to?

Answer 17

dA = dsydθ, where ds is an element along the line revolved, y is the height of the line, and dθ is the angle of revolution.

Question 18

How do you find the total surface area of revolution if it is revolved by 2π radians?

Answer 18

A = integral from s1 to s2 of 2πyds, where s1 and s2 are the start and end points of the line/curve

Question 19

What is the equation for the centroid of y, the function we are revolving?

Answer 19

y(centroid) = integral of y*ds / integral of ds

Question 20

How do you get an equation for area using the centroid equation?

Answer 20

Multiply each side by 2π, and then you have A/S, then rearrange for A.

Question 21

What is Pappus’ 2nd Theorem?

Answer 21

A = 2π * y(centroid) * s