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M05.6 · Multiple integrals and change of variables

A free, illustrated module of 4 lessons and one interactive in M05 Multivariable and Matrix Calculus: Double integrals over rectangles; Regions of any shape and the order of integration; Polar coordinates and the Gaussian integral; Change of variables and the Jacobian determinant.

M05.6 L2 · Intermediate 4 lessons · about 100 min Live · in review

Part of M05 Multivariable and Matrix Calculus in the Mathematics track.

A double integral is a volume: fill the space under a surface with thin boxes, or cut it into slices and add their areas. These four lessons integrate over rectangles, over regions bounded by curves in either order, and over discs in polar coordinates, where they prove at last that the bell curve has area the square root of 2 pi. They end with the general change of variables, in which the Jacobian determinant rescales every cell, and with what that does to a probability density, the idea behind normalising flows.

What you need first. Riemann sums and the fundamental theorem of calculus (M04.5, M04.6), improper integrals (M04.7.3), functions of two variables (M05.1), and for the last lesson the Jacobian and its determinant as an area factor (M05.4.4). Joint densities (M06.9.4) help with the examples on probability but are not required.

Lessons

1. Double integrals over rectangles M05.6.1 25 min

   The volume under a surface over a rectangle is the limit of sums of thin boxes, and Fubini’s theorem computes it one slice at a time, in either order. Worked on x y and the running bowl over the unit square, checked with midpoint sums and scipy.integrate.dblquad.

2. Regions of any shape and the order of integration M05.6.2 25 min

   Integrate over a triangle, a quarter disc or a tetrahedron by reading the limits off a picture: the inner limits follow the edge of the region. Swap the order by redrawing the slices, compute probabilities of regions under a joint density, and check everything with dblquad and tplquad.

3. Polar coordinates and the Gaussian integral M05.6.3 25 min

   A polar cell has area r times dr times dtheta, so a disc is integrated with dA = r dr dtheta. That one factor r turns the Gaussian over the plane into an easy integral, 2 pi, and proves at last that the bell curve has area the square root of 2 pi.

4. Change of variables and the Jacobian determinant M05.6.4 25 min

   A smooth one-to-one map sends each small cell to a near-parallelogram whose area is |det J| times the cell’s. So integrals change variables with the factor |det J|, polar coordinates included, and a density pushed through a map is divided by it: the idea behind normalising flows.

The interactive

M05.6-X1 · Region integrator. Choose a function, a region and a grid of Cartesian or polar cells, then slide the number of cells. The cells are shaded by the function’s value, those that cross the boundary are marked, and the read-out sets the Riemann sum beside the exact integral. It lives in Polar coordinates and the Gaussian integral.