if-w-f-x-y-and-x-r-cos-y-rsin-then-prove-that-w-rr-w-0-

Question Number 164121 by mkam last updated on 14/Jan/22

i f w = f (x, y) a n d x = r c o s θ, y = r s i n θ

t h e n p r o v e t h a t w_{r r} + w_{θ θ} = 0 ?

Commented by mkam last updated on 14/Jan/22

? ? ? ?

Commented by mkam last updated on 15/Jan/22

? ? ?

Commented by mr W last updated on 15/Jan/22

y o u c a n t p r o v e!

{i t}^{'} s f a l s e!

Answered by mr W last updated on 15/Jan/22

w_{r} = f_{x} \cos θ + f_{y} \sin θ

w_{r r} = f_{x x} \cos^{2} θ + f_{x y} \cos θ \sin θ + f_{y x} \sin θ \cos θ + f_{y y} \sin^{2} θ

w_{θ} = - f_{x} r \sin θ + f_{y} r \cos θ

w_{θ θ} = - f_{x} r \cos θ - r \sin θ (- f_{x x} r \sin θ + f_{x y} r \cos θ) - f_{y} r \sin θ + r \cos θ (- f_{x y} r \sin θ + f_{y y} r \cos θ)

w_{θ θ} = - f_{x} r \cos θ + f_{x x} r^{2} \sin^{2} θ - f_{x y} r^{2} \sin θ \cos θ - f_{y} r \sin θ - f_{x y} r^{2} \sin θ \cos θ + f_{y y} r^{2} \cos^{2} θ

w_{θ θ} = - f_{x} r \cos θ - f_{y} r \sin θ + f_{x x} r^{2} \sin^{2} θ + f_{y y} r^{2} \cos^{2} θ - 2 f_{x y} r^{2} \sin θ \cos θ

w_{r r} + w_{θ θ} \neq 0

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