Mostre que w=f(u, v), onde u=x+a t e v=y+b t, satisfaz à equação
partial wpartial t=a partial wpartial x+b partial wpartial y

Passo 1

O enunciado define
w(x,y,t)=fl(u,vr),
onde
u = x + a,t, v = y + b,t,
e pede para mostrar que
partial wpartial t = a,partial wpartial x + b,partial wpartial y
O que precisamos é aplicar a regra da cadeia para derivadas parciais.

Passo 2

Para calcular partial w/partial t usamos a regra da cadeia:
partial wpartial t = partial fpartial u,partial upartial t + partial fpartial v,partial vpartial t
Como partial u/partial t=a e partial v/partial t=b, temos
partial wpartial t = a,partial fpartial u + b,partial fpartial v

Passo 3

Agora calculamos partial w/partial x e partial w/partial y. De novo, pela cadeia:
partial wpartial x = partial fpartial u,partial upartial x + partial fpartial v,partial vpartial x = partial fpartial ucdot1 + partial fpartial vcdot0 = partial fpartial u,
partial wpartial y = partial fpartial u,partial upartial y + partial fpartial v,partial vpartial y = 0 + partial fpartial vcdot1 = partial fpartial v

Passo 4

Basta combinar:
a,partial wpartial x + b,partial wpartial y = a,partial fpartial u + b,partial fpartial v
que coincide com o resultado de partial w/partial t. Logo, a equação é satisfeita.

Resposta

partial wpartial t = a,partial wpartial x + b,partial wpartial y

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