Exercise · ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-003 · ★☆☆ Basic
For polar coordinates \(x=r\cos\theta,\ y=r\sin\theta\), the Jacobian \(\frac{\partial(x,y)}{\partial(r,\theta)}\) is:
Choose one option.
Show answer
- A. \(r\)
Explanation
\(J=\begin{vmatrix}\cos\theta&-r\sin\theta\\\sin\theta&r\cos\theta\end{vmatrix}=r\cos^2\theta+r\sin^2\theta=r\). That is why \(dA=r\,dr\,d\theta\).
Review the concept: Change of variables in multiple integrals
A change of variables turns the integral of a function \(f\) over a region \(R\) of the \(xy\)-plane into one over a region \(S\) of the \(uv\)-plane, correcting the area with the Jacobian \(J\) of the transformation.
\[\begin{aligned}\iint_R f\,dx\,dy&=\iint_S f(x(u,v),y(u,v))\,|J|\,du\,dv\\J&=\frac{\partial(x,y)}{\partial(u,v)}=\begin{vmatrix}x_u&x_v\\y_u&y_v\end{vmatrix}\end{aligned}\]
More practice problems: change of variables in multiple integrals
- ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-001 For the change x=2u,\ y=3v, what is |J|=|(∂(x,y))/(∂(u,v))|?
- ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-002 For x=u+v,\ y=u-v, the value of |J| is:
- ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-004 If x=u^2-v^2,\ y=2uv, the Jacobian (∂(x,y))/(∂(u,v)) is:
- ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-005 With x=2u,\ y=3v, the area of the ellipse tfracx^24+tfracy^29≤1 is:
- ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-006 If u=x+y,\ v=x-y, then dx dy=
- ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-007 If at a point J=-3, what does the transformation do near it?
- ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-008 Let R be the region given by x=u+v,\ y=u-v with 0≤ u,v≤1. Compute iint_R (x^2-y^2) dA.
- ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-009 Let R be the triangle with vertices (0,0),(1,0),(0,1). With u=y-x,\ v=x+y, compute iint_R e^(y-x)/(…
← ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-002 ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-004 →