chapter 6

Question 1

What is a double integral?

Answer 1

I = ∬A f(x, y) dA where A is the region/domain of integration and dA is the infinitesimal area element.

Question 2

What is the area element in Cartesian coordinates?

Answer 2

dA = dx dy

Question 3

What is the area element in polar coordinates?

Answer 3

For (x, y) = (R cos φ, R sin φ): dA = R dR dφ

Question 4

What is the Jacobian determinant

Answer 4

J(u, v) = (∂x/∂u)(∂y/∂v) − (∂x/∂v)(∂y/∂u)

Question 5

How do you change variables in a double integral?

Answer 5

If x = x(u, v), y = y(u, v): ∬A f(x, y) dx dy = ∬S f(x(u, v), y(u, v)) |J(u, v)| du dv where region A in (x, y)-plane corresponds to region S in (u, v)-plane.