Consider z = f(x, y), where x = r cos theta and y = r sin theta. Use to Chain Ru

Question

Consider z = f(x, y), where x = r cos theta and y = r sin theta. Use to Chain Rule to show that partial differential z/partial differential r = cos theta partial differential z/partial differential x + sin theta partial differential z/partial differential y, partial differential z/partial differential theta = -r sin theta partial differential z/partial differential x + r cos theta partial differential z/partial differential y. Use the above results to show that partial differential z/partial differential x = cos theta partial differential z/partial differential r - 1/r sin theta partial differential z/partial differential theta, partial differential z/partial differential y = sin theta partial differential z/partial differential r + 1/r cos theta partial differential z/partial differential theta. Use (a) to show that partial differential^2 z/partial differential r^2 = cos^2 theta partial differential^2 z/partial differential x^2 + 2 sin theta cos theta partial differential^2 z/partial differential x partial differential y + sin^2 theta partial differential^2 z/partial differential y^2, 1/r^2 partial differential^2 z/partial differential theta^2 + 1/r partial differential z/partial differential r = sin^2 theta partial differential^2 z/partial differential x^2 - 2 sin theta cos theta partial differential^2 z/partial differential x partial differential y + cos^2 theta partial differential^2 z/partial differential y^2, and consequently partial differential^2 z/partial differential x^2 + partial differential^2 z/partial differential y^2 = partial differential^2/partial differential r^2 + 1/r partial differential z/partial differential r + 1/r^2 partial differential^2 z/partial differential theta^2

Explanation / Answer

(a) dz/dr = (dz/dx)*(dx/dr) + (dz/dy)*(dy/dr) =

(dz/dx)*cos(theta) + (dz/dy)*sin(theta), and

dz/d(theta) = (dz/dx)*(dx/d(theta)) + (dz/dy)*(dy/d(theta)) =

(dz/dx)*r*(-sin(theta)) + (dz/dy)*r*cos(theta) =

r*[-(dz/dx)*sin(theta) + (dz/dy)*cos(theta)].

(b) (dz/dr)^2 = [(dz/dx)*cos(theta) + (dz/dy)*sin(theta)]^2 =

(dz/dx)^2*cos^2(theta) + (dz/dy)^2*sin^2(theta) +

2*(dz/dx)*(dz/dy) *sin(theta)*cos(theta), and

(1/r^2)*(dz/d(theta))^2 =

(1/r^2)*r^2*[(dz/dx)^2*sin^2(theta) + (dz/dy)^2*cos^2(theta) -

2*(dz/dx)*dz/dy)* sin(theta)*cos(theta)].

Thus (dz/dr)^2 + (1/r^2)*(dz/d(theta))^2 =

[(dz/dx)^2 + (dz/dy)^2] * (sin^2(theta) + cos^2(theta)) =

(dz/dx)^2 + (dz/dy)^2 as expected.

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