Paper 3, Section II, C

Vector Calculus | Part IA, 2018

Given a one-to-one mapping u=u(x,y)u=u(x, y) and v=v(x,y)v=v(x, y) between the region DD in the (x,y)(x, y)-plane and the region DD^{\prime} in the (u,v)(u, v)-plane, state the formula for transforming the integral Df(x,y)dxdy\iint_{D} f(x, y) d x d y into an integral over DD^{\prime}, with the Jacobian expressed explicitly in terms of the partial derivatives of uu and vv.

Let DD be the region x2+y21,y0x^{2}+y^{2} \leqslant 1, y \geqslant 0 and consider the change of variables u=x+yu=x+y and v=x2+y2v=x^{2}+y^{2}. Sketch DD, the curves of constant uu and the curves of constant vv in the (x,y)(x, y)-plane. Find and sketch the image DD^{\prime} of DD in the (u,v)(u, v)-plane.

Calculate I=D(x+y)dxdyI=\iint_{D}(x+y) d x d y using this change of variables. Check your answer by calculating II directly.

Typos? Please submit corrections to this page on GitHub.