Calcule o jacobiano da transformação dada por
\begin{aligned} & x=x(r, \theta, \varphi)=r \operatorname{sen}(\varphi) \cos (\theta) \\\ & y=y(r, \theta, \varphi)=r \operatorname{sen}(\varphi) \operatorname{sen}(\theta) \\\ & z=z(r, \theta, \varphi)=r \cos (\varphi) \end{aligned}

Passo 1

O enunciado nos dá a transformação de coordenadas esféricas (r,\theta,\varphi) para cartesianas (x,y,z):
x = r\,\sin(\varphi)\cos(\theta),\quad y = r\,\sin(\varphi)\sin(\theta),\quad z = r\,\cos(\varphi).
O objetivo é calcular o Jacobiano, ou seja o determinante da matriz das derivadas parciais \displaystyle \frac{\partial(x,y,z)}{\partial(r,\theta,\varphi)}.

Passo 2

Primeiro montamos a matriz jacobiana J = \bigl[\partial(x,y,z)/\partial(r,\theta,\varphi)\bigr]. Calculando cada derivada:
\frac{\partial x}{\partial r}=\sin\varphi\cos\theta,\quad \frac{\partial x}{\partial \theta}=-r\sin\varphi\sin\theta,\quad \frac{\partial x}{\partial \varphi}=r\cos\varphi\cos\theta,
\frac{\partial y}{\partial r}=\sin\varphi\sin\theta,\quad \frac{\partial y}{\partial \theta}=r\sin\varphi\cos\theta,\quad \frac{\partial y}{\partial \varphi}=r\cos\varphi\sin\theta,
\frac{\partial z}{\partial r}=\cos\varphi,\quad \frac{\partial z}{\partial \theta}=0,\quad \frac{\partial z}{\partial \varphi}=-r\sin\varphi.
Logo
J = \begin{pmatrix} \sin\varphi\cos\theta & -r\sin\varphi\sin\theta & r\cos\varphi\cos\theta\\\ \sin\varphi\sin\theta & \;r\sin\varphi\cos\theta & r\cos\varphi\sin\theta\\\ \cos\varphi & 0 & -r\sin\varphi \end{pmatrix}.

Passo 3

Para achar o determinante, usamos a expansão por co-fatores:
\det J = x_{r}\,(y_{\theta}z_{\varphi}-y_{\varphi}z_{\theta}) - x_{\theta}\,(y_{r}z_{\varphi}-y_{\varphi}z_{r}) + x_{\varphi}\,(y_{r}z_{\theta}-y_{\theta}z_{r}).
Substituindo as derivadas e simplificando term a term, obtemos
\det J = -\,r^{2}\sin\varphi\Bigl(\cos^{2}\theta+\sin^{2}\theta\Bigr) = -\,r^{2}\sin\varphi.

Passo 4

Como em integrais de volume usamos o valor absoluto do Jacobiano, temos
\bigl|\det J\bigr| = r^{2}\sin\varphi.

Resposta

r^{2}\sin(\varphi)

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