Change of Variables and the Jacobian
Change coordinates in a multiple integral, and see the Jacobian determinant as the local factor by which area or volume is stretched.
What a learner can do afterwards
- Compute a Jacobian and use it to transform a double integral to easier coordinates
- Recover the polar, cylindrical and spherical area elements as Jacobians rather than as memorised facts
- Design a substitution that turns an awkward region into a rectangle
1 · Read
A change of variables swaps old coordinates for new ones, the way u substitution swaps x for u. Each tiny rectangle in the new grid lands stretched in the old grid. The Jacobian, written J, is the determinant of the matrix of first partial derivatives. It measures that local stretching. The area element becomes dx dy equals the absolute value of J times du dv. The absolute value matters because area stays positive even when the determinant comes out negative.
Work in three moves. First pick a map that straightens the region, so each awkward boundary becomes u equals a constant or v equals a constant. Then rewrite the integrand and the limits in the new variables. Finally multiply by the absolute value of the Jacobian. Polar coordinates are the most used case: x equals r cos t and y equals r sin t, with Jacobian r. That r comes from the determinant of the polar partials.
Take the map x equals 2u and y equals 3v. All partials are constant, and the determinant is 2 times 3 minus 0, which is 6. So one unit of uv area covers 6 units of xy area. The classic error is the polar cousin: writing the area element as dr dt and dropping the r. At radius 5 the Jacobian equals 5, so forgetting it throws away a factor of 5.
Choose polar when you see circular regions, radial symmetry, or x squared plus y squared in the integrand. Sketch the region to read off the angle sweep and the inner and outer radii. The same stretching idea gives r in cylindrical coordinates and rho squared sin phi in spherical ones. If the new region is no simpler than the old one, try a different map.
You pick a map that straightens the region, rewrite everything, and multiply by the absolute Jacobian that measures the local stretch.
More to read
5.7 Change of Variables in Multiple Integrals
Substitution in higher dimensions works like u-substitution with a correction factor attached. A map from uv variables to xy variables stretches each tiny rectangle by the absolute value of the Jacobian determinant, so dx dy becomes |J| du dv. The method has three moves: pick a map that straightens the region, rewrite the integrand and limits, and multiply by the Jacobian. Regions with slanted or curved sides often turn into plain rectangles. Design the map from the boundary curves: each awkward boundary should become u = constant or v = constant. Then compute the partials, take the determinant, and do not forget its absolute value. Check the orientation sign separately, because area elements stay positive even when the determinant comes out negative.
5.3 Double Integrals in Polar Coordinates
Polar coordinates are the most used change of variables: x = r cos t, y = r sin t, with Jacobian determinant r. That single letter r comes from the determinant of the polar partials. Circular regions, radial symmetry and integrands with x^2 + y^2 all signal polar. Limits become r ranges and angle ranges instead of x and y boxes. Set up carefully: the area element is r dr dt, and forgetting the r is the classic error. Sketch the region to read off the angle sweep and the inner and outer radii. The same pattern extends to cylindrical and spherical coordinates, where the Jacobian factors r and rho^2 sin(phi) play the identical stretching role.
5.1 Double Integrals over Rectangular Regions
Before changing variables, be fluent with plain double integrals over rectangles. A double integral adds up values over a region, and over a rectangle it splits into two iterated single integrals. Fubini's theorem says the order rarely matters for continuous functions, so pick whichever inner integral looks easier. Volume under a surface is the guiding picture. Rectangles are also the target of every good substitution: the whole point of a change of variables is to turn an awkward region into one of these. Compare the original limits with the transformed ones to see what the map bought you. If the new region is not simpler, try a different map.
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Where it sits
14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.