Compute Integral Using Jacobian Det in Polar Coordinates

[Mamooie312]

Homework Statement

Determine the Jacobian determinant for "polar" coordinates and use that to compute the intergral . . . Blah blah blah that's not the point.

Homework Equations

(x,y) maps by T to (r, theta) or (theta, r) detT = jacobian

The Attempt at a Solution

Anyways, first I treated it as a map from (x,y) to (theta, r) and I got the answer in the book but negative and then I did it the other way and I got the answer in the book. Why is this and if I differ how I map the function then I supposed to integrate the opposite way (like from pi to 0 instead of from 0 to pi)

 

[HallsofIvy]

I have no idea what you are saying or what question you are asking. Please show exactly what you did in calculating the Jacobian.

 

[Mamooie312]

T: (x,y) to (theta, r) x = rcostheta y = rsintheta derivative matrix of T: t11 = -rsintheta t12 = costheta t21 = rcostheta t22 = sintheta detT = -r(sintheta)(squared) - r(costheta)(squared) = -r but the Jacobian should just be r and if you map T: (x,y) to (r, theta) instead of what I did then it works out to be r, why is this.

 

[Liquidxlax]

T: (x,y) to (theta, r) x = rcostheta y = rsintheta derivative matrix of T: t11 = -rsintheta t12 = costheta t21 = rcostheta t22 = sintheta detT = -r(sintheta)(squared) - r(costheta)(squared) = -r but the Jacobian should just be r and if you map T: (x,y) to (r, theta) instead of what I did then it works out to be r, why is this.

take your matrix and do the determinant.

rsin²(x) + rcos²(x) is what you'll get