Explicit proof of the Jacobian inverse

  • Thread starter Physgeek64
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Homework Statement


Given the transformations ##x^2+y^2=2*r*cos(theta)## and ##x*y=r*sin(theta)## prove the Jacobian explicitly

The question then goes on to ask how r and theta are related to the cylindrical coordinates rho and phi. I think ##r=1/2*(x^2+y^2)## and hence ##r=1/2 rho## but Im not really sure about this part I'm afraid, so haven't gotten very far

Homework Equations


##(partial(x,y)/partial(r,theta))*partial(r,theta)/partial(x,y))=1##

The Attempt at a Solution


So by multiplying the second expression by two and then squaring both and adding we get ##r=1/2*(x^2+y^2)## from which we can find the first two elements in the second Jacobian to be x and y respectively. By dividing the two transformations we can also get an expression for theta, which is relativity simple to differentiate, however when isolating x and y I seem to be going into pages of algebra, which leads me to think I have got the wrong approach, and there must be a simpler way since this is meant to be a relatively quick question.

Many thanks
 

Answers and Replies

  • #2
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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 

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