Changing variables in multiple integrals

In summary, the conversation discusses dividing a region into small rectangles in the x,y plane and finding the area element in the u,v system. The process involves using partial derivatives to approximate changes in u and v at each edge of the rectangle, and visualizing the rectangle as a parallelogram in the u,v plane to calculate its area using a determinant.
  • #1
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TL;DR Summary
I want to understand intuition behind the Jacobian matrix proof
Suppose we have a region R in the x-y plane and divide the region into small rectangles of area dxdy. If the integrand or the limits of integration were to be simplified with the introduction of new variables u and v instead of x and y, how can I supply the area element in the u-v system in the same x-y plane?
To generalise let u and v are both functions of x and y. By Holding y=y0 constant and changing x from x0 to x0+Δx this would result in changes in u and v.
Δu1≈ (∂u/∂x)Δx The partial derivatives evaluated at (x0,y0)
Δv1≈ (∂v/∂x)Δx
will serve as a good approximation to u and v at (x0+Δx,y0)

Similarly, holding x=x0+Δx and changing y from y0 to y0+Δy this will result in changes
Δu2≈ (∂u/∂y)Δy
Δv2≈ (∂v/∂y)Δy The partial derivatives evaluated at (x0+Δx,y0)

Holding y=y0+Δy and changing x from x+Δx to x0 will result in changes
Δu3≈ -(∂u/∂x)Δx
Δv3≈ -(∂v/∂x)Δx The partial derivatives evaluated at (x0+Δx,y0+Δy)

Holding x=x0 and changing y from y0+Δy to y0 will result in changes
Δu4≈ -(∂u/∂y)Δy
Δv4≈ -(∂v/∂y)Δy The partial derivatives evaluated at (x0,y0+Δy)

What I did was look at how u and v changed with each edge of the rectangle. But I cannot seem to get how the above fits together into forming the area in the uv system
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  • #2
I suggest visualizing a rectangle in the x,y plane with sides given by two vectors mapping to a parallelogram in the u,v plane with sides defined by two vectors.

In the x,y plane the vectors are ##(0,\triangle x)## and ##(\triangle y, 0)##.

The area of a parallelogram is given by a determinant. https://mathinsight.org/relationship_determinants_area_volume
 

Related to Changing variables in multiple integrals

1. What is the purpose of changing variables in multiple integrals?

Changing variables in multiple integrals is a technique used to simplify the integrand and make the integration process easier. It can also help to visualize and understand the geometric interpretation of the integral.

2. How do you change variables in double integrals?

To change variables in double integrals, we use the substitution method. This involves replacing the original variables with new ones and then rewriting the integral in terms of the new variables. The new variables are chosen in such a way that the integral becomes easier to evaluate.

3. Can changing variables affect the value of a multiple integral?

Yes, changing variables can affect the value of a multiple integral. This is because the limits of integration and the integrand may change when we substitute new variables. It is important to keep track of these changes and adjust the integral accordingly.

4. Are there any specific guidelines for choosing new variables in multiple integrals?

There are no specific guidelines for choosing new variables in multiple integrals. However, it is helpful to choose variables that simplify the integrand and make the integral easier to evaluate. It is also important to choose variables that preserve the orientation and boundaries of the original integral.

5. Can changing variables be used in triple integrals as well?

Yes, changing variables can be used in triple integrals as well. The same substitution method is used, but we need to introduce a third variable and make sure that the new variables preserve the orientation and boundaries of the original triple integral.

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