Calculando a derivada parcial em relação a
v
teremos,
(del z)/(del v) = (del z ( x , y ))/(del x) . (del x)/(del v) + (del z ( x , y ))/(del y) . (del y)/(del v)
= - (del z ( x , y ))/(del x) . e^u sen ( v ) + (del z ( x , y ))/(del y) e^u cos ( v )
Assim,
(del^2 z)/(del v^2) = (del x)/(del v) [- (del z ( x , y ))/(del x) . e^u sen ( v )] + (del x)/(del v) [(del z ( x , y ))/(del y) e^u cos ( v )]
Pela regra do produto,
= - (del x)/(del v) [(del z (x , y))/(del x)] e^u sen (v) - [(del z (x , y))/(del x) e^u cos (v)]
+ (del y)/(del v) [(del z (x , y))/(del y)] e^u cos (v) - [(del z (x , y))/(del y) e^u sen (v)]
Fazendo separadamente teremos,
(del x)/(del u) [(del z (x , y))/(del x)] = (del^2 z (x , y))/(del x^2) (del x)/(del v) + (del^2 z (x , y))/(del y del x) (del y)/(del v)
= - (del^2 z (x , y))/(del x^2) e^u sen ( v ) + (del^2 z (x , y))/(del y del x) e^u cos ( v )
Da mesma forma,
(del y)/(del u) [(del z (x , y))/(del y)] = (del^2 z (x , y))/(del x del y) (del x)/(del v) + (del^2 z (x , y))/(del y^2) (del y)/(del v)
= - (del^2 z (x , y))/(del x del y) e^u sen ( v ) + (del^2 z (x , y))/(del y^2) e^u cos ( v )
Logo,
(del^2 z)/(del u^2) = - [- (del^2 z (x , y))/(del x^2) e^u sen ( v ) + (del^2 z (x , y))/(del y del x) e^u cos ( v )] e^u sen (v) - [(del z (x , y))/(del x) e^u cos (v)]
+ [- (del^2 z (x , y))/(del x del y) e^u sen ( v ) + (del^2 z (x , y))/(del y^2) e^u cos ( v )] e^u cos (v) - [(del z (x , y))/(del y) e^u sen (v)]
= [(del^2 z (x , y))/(del x^2) e^(2 u) sen^2 (v) - (del^2 z (x , y))/(del y del x) e^(2 u) cos (v) s e n ( v )] - [(del z (x , y))/(del x) e^u cos (v)]
= [- (del^2 z (x , y))/(del x del y) e^(2 u) sen (v) c o s ( v ) + (del^2 z (x , y))/(del y^2) e^(2 u) cos^2 ( v )] - [(del z (x , y))/(del y) e^u sen (v)]
Utilizando o Teorema de Schwarz
= [(del^2 z (x , y))/(del x^2) e^(2 u) sen^2 (v) - (del^2 z (x , y))/(del x del y) e^(2 u) cos (v) s e n ( v )] - [(del z (x , y))/(del x) e^u cos (v)]
= [- (del^2 z (x , y))/(del x del y) e^(2 u) c o s ( v ) sen (v) + (del^2 z (x , y))/(del y^2) e^(2 u) cos^2 ( v )] - [(del z (x , y))/(del y) e^u sen (v)]
Logo,
= [(del^2 z (x , y))/(del x^2) e^(2 u) sen^2 (v) - 2 (del^2 z (x , y))/(del x del y) e^(2 u) cos (v) s e n (v) + (del^2 z (x , y))/(del y^2) e^(2 u) cos^2 ( v )] - [(del z (x , y))/(del x) e^u cos (v)] - [(del z (x , y))/(del y) e^u sen (v)]