Prove the change of variables formula for double integrals

In summary: S is the region in the uv-plane that corresponds to the region that corresponds to the region R under the transformation given by x = g(u,v), y = h(u,v).U and v are related to x and y by the equation u^2 + v^2 = 1.The determinant of the Jacobian is the expression on the right side of the homework equation.
  • #1
alanthreonus
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Homework Statement


Use Green's Theorem to prove for the case f(x,y) = 1

[tex]\int\int_R dxdy = \int\int_S |\partial(x,y)/\partial(u,v)|dudv[/tex]

EDIT: R is the region in the xy-plane that corresponds to the region S in the uv-plane under the transformation given by x = g(u,v), y = h(u,v), and the expression on the right is the Jacobian.

Homework Equations


[tex]A = \oint _C xdy = -\oint _C ydx = 1/2\oint _C xdy - ydx[/tex]

The Attempt at a Solution


My textbook says that the left side of the equation I'm trying to prove is A(R), so I can apply the first part of the equation in 2., but I don't understand what it's talking about because the equation in 2. deals with line integrals, and I'm trying to prove an equation with double integrals.
 
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  • #2
alanthreonus said:

Homework Statement


Use Green's Theorem to prove for the case f(x,y) = 1

[tex]\int\int_R dxdy = \int\int_S |\partial(x,y)/\partial(u,v)|dudv[/tex]
What is R? S?
How are u and v related to x and y?
Isn't the expression in absolute values on the right side the determinant of the Jacobian?
alanthreonus said:

Homework Equations


[tex]A = \oint _C xdy = -\oint _C ydx = 1/2\oint _C xdy - ydx[/tex]


The Attempt at a Solution


My textbook says that the left side of the equation I'm trying to prove is A(R), so I can apply the first part of the equation in 2., but I don't understand what it's talking about because the equation in 2. deals with line integrals, and I'm trying to prove an equation with double integrals.
 
  • #3
R is the region in the xy-plane that corresponds to the region S in the uv-plane under the transformation given by x = g(u,v), y = h(u,v), and, yes, that is the Jacobian.
 

1. What is the change of variables formula for double integrals?

The change of variables formula for double integrals is a mathematical formula that allows us to express a double integral in terms of a new set of variables. This is useful when the original variables are difficult to work with or when we want to integrate over a different region.

2. Why do we need the change of variables formula for double integrals?

The change of variables formula is necessary because it allows us to solve more complex integrals by transforming them into simpler ones. It also helps us to solve integrals over non-rectangular regions, which would be difficult to do using traditional integration methods.

3. How is the change of variables formula derived?

The change of variables formula is derived from the multivariable chain rule. By substituting the new variables into the original integral and using the chain rule, we can express the integral in terms of the new variables.

4. What are the steps for using the change of variables formula for double integrals?

The steps for using the change of variables formula are as follows:
1. Determine the appropriate transformation for the new variables.
2. Calculate the Jacobian of the transformation.
3. Substitute the new variables and the Jacobian into the original integral.
4. Evaluate the new integral in terms of the new variables.
5. If necessary, convert back to the original variables to get the final answer.

5. Can the change of variables formula be used for triple integrals?

Yes, the change of variables formula can be extended to triple integrals. The steps are similar to double integrals, but involve finding the Jacobian for the new variables in three dimensions. This allows us to integrate over more complex regions and solve more difficult integrals.

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