Change of Variables with Jacobians

[Solid Snake]

Homework Statement

Suggest a substitution/transformation that will simplify the following integral and find their jacobians:

$$\int\int_{R}x\sin(6x+7y)-3y\sin(6x+7y)dA$$

Homework Equations

[ tex ] \int\int_{R}x\sin(6x+7y)-3y\sin(6x+7y)dA [ / tex ]

The Attempt at a Solution

This topic is change of variables (with Jacobians). My book does a terrible job of explaining it to someone who isn't the most talented at reading small abstract concepts (as if it were written as a professional math paper), and coming to 10,000 conclusions based on it. I basically understand this to be the same as "u-substitution" that I did in one variable calculus, except obviously this is for two variables. So I tried this as follows:

u=x*sin(6x+7y)
v=3y*sin(6x+7y)

From here I'm beyond lost. I mean, I know the definition of the jacobian, but I don't see any function x here (it requires you to get the derivative of x in terms of both u and v, and the derivative of y in terms of u and v). I don't even know if these substitutions are even correct. Honestly I'm beyond lost, and I would love it if someone could just explain this to me in English (something many math professors have a hard time doing). Can someone explain to me how one goes about finding the correct substitution or transformation when presented with such integrals (or ANY integral in general)? How does one find the transformations??

EDIT: I also thought of factoring out that sin(6x+7y) and then setting u=6x+7y and v=x-3y. I'm beyond lost...

 

[LCKurtz]

Homework Statement

Suggest a substitution/transformation that will simplify the following integral and find their jacobians:

$$\int\int_{R}x\sin(6x+7y)-3y\sin(6x+7y)dA$$

Homework Equations

[ tex ] \int\int_{R}x\sin(6x+7y)-3y\sin(6x+7y)dA [ / tex ]

The Attempt at a Solution

This topic is change of variables (with Jacobians). My book does a terrible job of explaining it to someone who isn't the most talented at reading small abstract concepts (as if it were written as a professional math paper), and coming to 10,000 conclusions based on it. I basically understand this to be the same as "u-substitution" that I did in one variable calculus, except obviously this is for two variables. So I tried this as follows:

u=x*sin(6x+7y)
v=3y*sin(6x+7y)

From here I'm beyond lost. I mean, I know the definition of the jacobian, but I don't see any function x here (it requires you to get the derivative of x in terms of both u and v, and the derivative of y in terms of u and v). I don't even know if these substitutions are even correct. Honestly I'm beyond lost, and I would love it if someone could just explain this to me in English (something many math professors have a hard time doing). Can someone explain to me how one goes about finding the correct substitution or transformation when presented with such integrals (or ANY integral in general)? How does one find the transformations??

EDIT: I also thought of factoring out that sin(6x+7y) and then setting u=6x+7y and v=x-3y. I'm beyond lost...

I like that last idea. Why are you lost? All you have to do is calculate the Jacobian.