Double integral change of variable

In summary, the conversation discusses a problem involving solving a double integral with constraints on a unit circle. The problem applies a change of variables using u and v, but the resulting equation seems odd and may not be the best approach. The conversation also mentions the use of Jacobian and the transformation of the integral. There is also a mention of the unit disk and a mapping of complex numbers.
  • #1
philnow
83
0

Homework Statement



Hey all. The problem is to solve the double integral xy da where the constraints C is x^2 + y^2 = 1, with the change of variables x = u^2 - v^2 and y = 2uv

The problem is applying the change of variables to the constraint unit circle. After the algebra I end up with (u2+v2)^2 = 1. What shape does this represent? How can I find the constraints for U and V given this... odd... equation?
 
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  • #2
I suppose you mean the domain to by [itex]x^2 + y^2 \leq 1[/itex], the interior of the circle. That substitution seems strange to me for this problem too. Your equation (u2+v2)2 = 1 is equivalent to (u2+v2) = 1, and the interior of the xy circle maps to the interior of the uv circle. Throw in the Jacobian and it seems like the transformed integral is worse than the original. I am a bit curious where the problem came from and why that substitution is suggested.
 
  • #3
If (u^2+v^2)^2=1 then u^2+v^2=1. It looks to me like the u,v domain is still the unit disk. In terms of complex numbers it's just the mapping f(z)=z^2. But you don't have to know that.
 

Related to Double integral change of variable

What is a double integral change of variable?

A double integral change of variable is a mathematical technique used to simplify the evaluation of a double integral by transforming it into a new coordinate system. This can make the integral easier to solve and can also provide insight into the geometric properties of the region of integration.

How do you perform a double integral change of variable?

To perform a double integral change of variable, you must first choose an appropriate transformation that will simplify the integral. This transformation must be one-to-one and continuously differentiable. Then, you must compute the Jacobian of the transformation, which is a matrix that relates the original variables to the new variables. Finally, you substitute the new variables into the original integral and evaluate it using the limits and integrand in terms of the new variables.

What are the advantages of using a double integral change of variable?

Using a double integral change of variable can simplify the evaluation of integrals, making them easier to solve. It can also provide geometric insight into the region of integration and can be used to solve integrals that would otherwise be difficult or impossible to solve.

Are there any limitations to using a double integral change of variable?

Yes, there are some limitations to using a double integral change of variable. The transformation must be one-to-one and continuously differentiable, and the Jacobian must not be equal to zero within the region of integration. Additionally, the transformation must be able to map the region of integration onto a simpler shape, otherwise the integral may not be simplified.

What are some common transformations used in double integral change of variable?

Some common transformations used in double integral change of variable include polar coordinates, cylindrical coordinates, and spherical coordinates. These transformations are often useful for evaluating integrals over circular, cylindrical, or spherical regions, respectively. Other transformations, such as linear transformations, may also be used depending on the specific integral being evaluated.

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