Generalized change of variables?

In summary, the concept of changing variables is fundamental in mathematics and allows us to simplify complicated situations by changing our perspective. This applies not only to integration, but also to other mathematical objects. The change of variables theorem generalizes to situations involving two differentiable manifolds and a one-to-one, differentiable transformation between them. The regular change of variables theorem can be thought of in terms of this, where the Jacobian represents the "distortion" caused by the derivative of the transformation. Overall, changing variables is a crucial concept in mathematics and can be seen as applying linear algebra to calculus.
  • #1
0rthodontist
Science Advisor
1,231
0
Does change of variables generalize to situations other than integration?
 
Mathematics news on Phys.org
  • #2
0rthodontist said:
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.
 
  • #3
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?
 
  • #4
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.
 
  • #5
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.
 

1. What is a generalized change of variables?

A generalized change of variables is a mathematical technique used to transform an integral from one coordinate system to another. It is commonly used in multivariable calculus and other areas of mathematics to simplify integrals and make them easier to solve.

2. How is a generalized change of variables performed?

To perform a generalized change of variables, you first need to determine the transformation function that will convert the variables in the original integral to the new variables. Then, you need to calculate the Jacobian of the transformation, which is a matrix of partial derivatives. Finally, you can substitute the new variables and the Jacobian into the original integral.

3. What is the purpose of a generalized change of variables?

The purpose of a generalized change of variables is to simplify integrals by converting them to a different coordinate system. This can make the integral easier to solve or evaluate, and it can also reveal underlying patterns or relationships between variables.

4. When is a generalized change of variables necessary?

A generalized change of variables is necessary when the original integral is difficult or impossible to solve in its current form. It can also be used to transform integrals into a more familiar or standard form, making it easier to apply known integration techniques.

5. What are some applications of generalized change of variables?

Generalized change of variables has various applications in mathematics, physics, and engineering. It is commonly used in solving differential equations, calculating volumes and surface areas in multivariable calculus, and in the study of geometric transformations and coordinate systems.

Similar threads

B Choosing which variable is the dependent or independent variable
    • General Math
Replies
16
Views
1K
I Change of variables; why do we take the absolute value?
    • General Math
Replies
6
Views
3K
B Can We Simplify the Taylor Series Expansion for e^(f(x,y))?
    • General Math
Replies
3
Views
664
B What is the notation for denoting constants in a function?
    • General Math
Replies
2
Views
741
B Can I replace ##X_n = i## with ##A## to type less? Rules of math.
    • General Math
Replies
5
Views
352
I 2nd order Taylor Series for a function in 3 or more variables?
    • General Math
Replies
4
Views
2K
I Solving Recursive Equations: Logistic Equation
    • General Math
Replies
19
Views
2K
I Area of a Circle: Solving the Equation
    • General Math
Replies
3
Views
785
I A matter of style? Algebra, Arithmetic, Variables
    • General Math
Replies
6
Views
1K
A Array variable of envelope function (parameter representation)
    • General Math
Replies
2
Views
907

Hot Threads

Recent Insights

Back
Top