1. Suppose that u = u(x,y), x(r, 0) = r cos 0, y(r, 0) = r...

 

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1. Suppose that u = u(x,y), x(r, 0) = r cos 0, y(r, 0) = r sin 0, and uxy = Uyx- (a) (b) (c) Use the Multivariable Chain Rule to prove that: ur = cos ux + sini uy Use part (a) to prove that: Urr = cos uxx + 2 sin 0 cos 0 uxy + sin uyy (Hint: Assume that Uxr = urx and Uyr = ury.) Use the Multivariable Chain Rule to prove that: ug = -r sin eux + rcose uy (d) (e) Use part (c) to prove that: - ue = r sin 0 uxx + r cos 0 uyy 2r sin cos 0 uxy - r cos eux-r sine uy (Hint: Assume that uxe = uex and uye = uey.) Use parts (a), (b), and (d) to prove that: 1 1 Urr +-Up + = r 12 uge uxx + uyy 1. Suppose that u = u(x,y), x(r, 0) = r cos 0, y(r, 0) = r sin 0, and uxy = Uyx- (a) (b) (c) Use the Multivariable Chain Rule to prove that: ur = cos ux + sini uy Use part (a) to prove that: Urr = cos uxx + 2 sin 0 cos 0 uxy + sin uyy (Hint: Assume that Uxr = urx and Uyr = ury.) Use the Multivariable Chain Rule to prove that: ug = -r sin eux + rcose uy (d) (e) Use part (c) to prove that: - ue = r sin 0 uxx + r cos 0 uyy 2r sin cos 0 uxy - r cos eux-r sine uy (Hint: Assume that uxe = uex and uye = uey.) Use parts (a), (b), and (d) to prove that: 1 1 Urr +-Up + = r 12 uge uxx + uyy