1. Usando a regra da cadeia para z=f(x, y) e w=f(x, y, z), calcule: \frac{\partial z}{\partial t}, \frac{\partial z}{\partial s} e \frac{\partial w}{\partial t}, \frac{\partial w}{\partial s} e \frac{\partial w}{\partial r}.
    (i) w=x^{2}+y^{2}+z^{2}, x=r \cos (s), y=r \operatorname{sen}(t) \operatorname{sen}(s), z=r \cos (t).

Passo 1

Temos a transformação de coordenadas
x = r\cos(s),\quad y = r\sin(t)\sin(s),\quad z = r\cos(t)
e a função
w = x^{2} + y^{2} + z^{2}.
Vamos usar a regra da cadeia para obter \frac{\partial z}{\partial t}, \frac{\partial z}{\partial s} e depois, usando \frac{\partial w}{\partial x}=2x, \frac{\partial w}{\partial y}=2y, \frac{\partial w}{\partial z}=2z, calcular \frac{\partial w}{\partial t}, \frac{\partial w}{\partial s} e \frac{\partial w}{\partial r}.

Passo 2

Calcule \frac{\partial z}{\partial t} e \frac{\partial z}{\partial s}.
Como z = r\cos(t), temos
\frac{\partial z}{\partial t} = r\cdot\bigl(-\sin(t)\bigr) = -\,r\sin(t),
e como z não depende de s,
\frac{\partial z}{\partial s} = 0.

Passo 3

Calcule \frac{\partial w}{\partial t} pela regra da cadeia
\frac{\partial w}{\partial t} = \frac{\partial w}{\partial x}\,\frac{\partial x}{\partial t} + \frac{\partial w}{\partial y}\,\frac{\partial y}{\partial t} + \frac{\partial w}{\partial z}\,\frac{\partial z}{\partial t}.
Aqui \frac{\partial x}{\partial t}=0, \frac{\partial y}{\partial t}=r\cos(t)\sin(s), \frac{\partial z}{\partial t}=-r\sin(t),
e \frac{\partial w}{\partial x}=2x, \frac{\partial w}{\partial y}=2y, \frac{\partial w}{\partial z}=2z.
Logo
\frac{\partial w}{\partial t} = 2x\cdot0 + 2y\bigl(r\cos(t)\sin(s)\bigr) + 2z\bigl(-r\sin(t)\bigr) = 2r\,y\cos(t)\sin(s)\;-\;2r\,z\sin(t).

Passo 4

Calcule \frac{\partial w}{\partial s} pela mesma ideia: \frac{\partial x}{\partial s}=-r\sin(s), \frac{\partial y}{\partial s}=r\sin(t)\cos(s), \frac{\partial z}{\partial s}=0.
Então
\frac{\partial w}{\partial s} = 2x\bigl(-r\sin(s)\bigr) + 2y\bigl(r\sin(t)\cos(s)\bigr) + 2z\cdot0 = -2r\,x\sin(s)\;+\;2r\,y\sin(t)\cos(s).

Passo 5

Calcule \frac{\partial w}{\partial r}: \frac{\partial x}{\partial r}=\cos(s), \frac{\partial y}{\partial r}=\sin(t)\sin(s), \frac{\partial z}{\partial r}=\cos(t).
Assim
\frac{\partial w}{\partial r} = 2x\cos(s) + 2y\sin(t)\sin(s) + 2z\cos(t).

Resposta

\displaystyle \frac{\partial z}{\partial t} = -\,r\sin(t),\quad \frac{\partial z}{\partial s}=0. \displaystyle \frac{\partial w}{\partial t} = 2r\,y\cos(t)\sin(s)\;-\;2r\,z\sin(t), \displaystyle \frac{\partial w}{\partial s} = -2r\,x\sin(s)\;+\;2r\,y\sin(t)\cos(s), \displaystyle \frac{\partial w}{\partial r} = 2x\cos(s) + 2y\sin(t)\sin(s) + 2z\cos(t).

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