Does change of variables generalize to situations other than integration?
That's really what mathematics is all about! Coordinate systems give us a way of simplifying complicated situations but a "real world problem" doesn't have a coordinate system attached- the particular coordinate system used is our decision. One of the most fundamental concepts in mathematics is changing from one coordinate system to the other- changing from one way of looking at a problem to another. That's what really happens in changing variables- we are changing from one coordinate system to another.
Okay, but I mean, does the change of variables theorem generalize to situations other than integration? I mean abstractly, from a logical standpoint. What are the minimum conditions you need to have something analogous to the change of variables theorem?
Sure it generalizes (this was just asked on my final exam)
Basically if you have two differentiable manifolds, and a one-to-one, differentiable transformation (with differentiable inverse, which is called a diffeomorphism) between them, then that diffeomorphism is sort of the generalized "change of variables" in a very rough sense.
The regular change of variables theorem can then be thought of in terms of that, where the Jacobian describes the "distortion" as the derivative of the diffeomorphism.
It's like my professor said, calc 3 is mostly linear algebra applied to calc 1 & 2.
it is hard to know what you are asking, but changing variables is basic to describing any mathematical object, and the change of variables theorem in integration just tells you how the volume of an object changes under a differentiable mapping.
basically it says that you know how the volume of a block changes under a linear transformation, namely by the determinant, so for a non linear mapping it changes locally by the determinant of the derivative.
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