HTSEFP: Multivariable calculus

Basic multivariable calculus. Should be mostly review from Calculus 3 or A/B.

1Reversing the order of integration¶

A seemingly impossible integral involving e, and a simple region in two dimensions.

1.1General solution¶

Have to change the order of integration. After that it should be a breeze.

1.2Examples¶

• Winter 2010 final, question 1

2Change of variables over a region¶

Notes: For the Jacobian: new variables go on the bottom, but take the inverse of that because otherwise it's hard to compute (so 1 over the value). Also positive (since, abs value etc)

2.1Examples¶

• Assignment 1, questions 1-3
• Winter 2006 final, question 3
• Winter 2010 final, question 2

3Deriving surface area in a specific coordinate system¶

Won't need to do this (no proofs etc)

3.1Examples¶

• Assignment 1, question 4

4Integrating using spherical coordinates¶

Basically a surface integral. Parametrise it, take the cross-product of the two partial derivatives, use that as the integrand and integrate over the limits of the region.

4.1Examples¶

• Winter 2006 final, question 4