Question

Encontre o jacobiano, se x=rcos(teta), y=rseno(teta) e z= sen(teta)

Answer 1

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Calcular o Jacobiano da seguinte transformação:

$\left\{\!\begin{array}{l} \mathsf{x(r,\,\theta,\,h)=r\,cos\,\theta}\\\\ \mathsf{y(r,\,\theta,\,h)=r\,sen\,\theta}\\\\ \mathsf{z(r,\,\theta,\,h)=sen\,\theta} \end{array}\right.$


O Jacobiano desta transformação é dado por

$\large\begin{array}{l} \mathsf{Jac_{\phi}(r,\,\theta,\,h)}=\mathsf{det}\begin{bmatrix} \mathsf{\frac{\partial x}{\partial r}}&\mathsf{\frac{\partial x}{\partial \theta}}&\mathsf{\frac{\partial x}{\partial h}}\\\\ \mathsf{\frac{\partial y}{\partial r}}&\mathsf{\frac{\partial y}{\partial \theta}}&\mathsf{\frac{\partial y}{\partial h}}\\\\ \mathsf{\frac{\partial z}{\partial r}}&\mathsf{\frac{\partial z}{\partial \theta}}&\mathsf{\frac{\partial z}{\partial h}} \end{bmatrix} \end{array}$


$\large\begin{array}{l} \mathsf{Jac_{\phi}(r,\,\theta,\,h)}=\mathsf{det}\begin{bmatrix} \mathsf{\frac{\partial}{\partial r}(r\,cos\,\theta)}&\mathsf{\frac{\partial}{\partial \theta}(r\,cos\,\theta)}&\mathsf{\frac{\partial}{\partial h}(r\,cos\,\theta)}\\\\ \mathsf{\frac{\partial}{\partial r}(r\,sen\,\theta)}&\mathsf{\frac{\partial}{\partial \theta}(r\,sen\,\theta)}&\mathsf{\frac{\partial}{\partial h}(r\,sen\,\theta)}\\\\ \mathsf{\frac{\partial}{\partial r}(sen\,\theta)}&\mathsf{\frac{\partial}{\partial \theta}(sen\,\theta)}&\mathsf{\frac{\partial}{\partial h}(sen\,\theta)} \end{bmatrix} \end{array}$


$\large\begin{array}{l} \mathsf{Jac_{\phi}(r,\,\theta,\,h)}=\mathsf{det}\begin{bmatrix} \mathsf{cos\,\theta}&\mathsf{-r\,sen\,\theta}&\mathsf{0}\\\\ \mathsf{sen\,\theta}&\mathsf{r\,cos\,\theta}&\mathsf{0}\\\\ \mathsf{0}&\mathsf{cos\,\theta}&\mathsf{0} \end{bmatrix} \end{array}$


Observe que a 3ª coluna matriz Jacobiana é toda nula. O determinante de uma matriz em que qualquer fila é nula sempre é igual a zero. Portanto,

$\large\begin{array}{l} \mathsf{Jac_{\phi}(r,\,\theta,\,h)=0\qquad\quad\checkmark}\end{array}$


Bons estudos! :-)