Sejam x=r cos theta, y=r sen theta e f(z)=u(x, y)+i v(x, y). Mostre que
partial upartial r & =partial upartial x cos theta+partial upartial y sen theta, & partial upartial theta & =-partial upartial x r sen theta+partial upartial y r cos theta e partial vpartial r & =partial vpartial x cos theta+partial vpartial y sen theta, & partial vpartial theta & =-partial vpartial x r sen theta+partial vpartial y r cos theta

Passo 1

O enunciado define as coordenadas polares
x = r \cos\theta e y = r \sin\theta e pede para encontrar as derivadas parciais de u(x,y) e v(x,y) em relação a r e \theta. O conteúdo é a aplicação da regra da cadeia para funções de múltiplas variáveis.

Passo 2

Para calcular partial upartial r e partial upartial theta, aplicamos a regra da cadeia em u(x(r,\theta),y(r,\theta)). A teoria diz que:
partial upartial r = partial upartial xpartial xpartial r + partial upartial ypartial ypartial r
partial upartial theta = partial upartial xpartial xpartial theta + partial upartial ypartial ypartial theta

Passo 3

Agora calculamos as derivadas de x e y em relação a r e \theta:
partial xpartial r = costheta, partial ypartial r = sintheta
partial xpartial theta = -rsintheta, partial ypartial theta = rcostheta

Passo 4

Substituindo esses valores nas fórmulas da regra da cadeia, obtemos para u:
partial upartial r = partial upartial xcostheta + partial upartial ysintheta
partial upartial theta = -partial upartial x,rsintheta + partial upartial y,rcostheta

Passo 5

O mesmo raciocínio vale para v(x,y): aplicamos exatamente as mesmas fórmulas trocando u por v e chegamos a:
partial vpartial r = partial vpartial xcostheta + partial vpartial ysintheta
partial vpartial theta = -partial vpartial x,rsintheta + partial vpartial y,rcostheta

Resposta

partial upartial r = partial upartial xcostheta + partial upartial ysintheta
partial upartial theta = -partial upartial x,rsintheta + partial upartial y,rcostheta
partial vpartial r = partial vpartial xcostheta + partial vpartial ysintheta
partial vpartial theta = -partial vpartial x,rsintheta + partial vpartial y,rcostheta

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