Càlcul de l’àrea de superfícies en coordenades cilíndriques

Escriu les fórmules pel càlcul de l’àrea d’una superfície domada en coordenades cilíndriques per: (a) \( r = g(\theta, z); \ h(z) = h(r, \theta). \) \begin{equation}\mathbf{e}(\theta, z) = (g(\theta, z) \cos \theta, g(\theta, z) \sin \theta, z)\end{equation}

\begin{equation}\mathbf{e}_\theta = \frac{\partial \mathbf{e}}{\partial \theta} = (g_\theta \cos \theta, g_\theta \sin \theta, 0) + g (-\sin \theta, \cos \theta, 0)\end{equation}\begin{equation}\mathbf{e}_z = \frac{\partial \mathbf{e}}{\partial z} = (g_z \cos \theta, g_z \sin \theta, 1)\end{equation}\begin{equation}\mathbf{e}_\theta \times \mathbf{e}_z = g_\theta \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\\cos \theta & \sin \theta & 0 \\g_z \cos \theta & g_z \sin \theta & 1\end{vmatrix} + g \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\-\sin \theta & \cos \theta & 0 \\g_z \sin \theta & g_z \cos \theta & 1\end{vmatrix}\end{equation}\begin{equation}= g_\theta (\sin \theta, -\cos \theta, 0) + g (\cos \theta, \sin \theta, -g_z)\end{equation}

suma vectors ortogonals

\begin{equation}\|\mathbf{e}_\theta \times \mathbf{e}_z\|^2 = g_\theta^2 + g^2 (1 + g_z^2)\end{equation}\begin{equation}\text{Àrea (S)} = \iint_D \sqrt{g_\theta^2(\theta, z) + g^2(\theta, z) (1 + g_z^2(\theta, z))} \, d\theta \, dz\end{equation}

(b) \( \mathbf{e}(r, \theta) = (r \cos \theta, r \sin \theta, h(r, \theta)) \)\begin{equation}\mathbf{e}_\theta = (-r \sin \theta, r \cos \theta, h_\theta)\end{equation}\begin{equation}\mathbf{e}_r = (\cos \theta, \sin \theta, h_r)\end{equation}\begin{equation}\mathbf{e}_\theta \times \mathbf{e}_r = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\-r \sin \theta & r \cos \theta & h_\theta \\\cos \theta & \sin \theta & h_r\end{vmatrix}\end{equation}\begin{equation}= (h_r \cos \theta – h_\theta \sin \theta) \mathbf{i} – (h_r \sin \theta + h_\theta \cos \theta) \mathbf{j} + r \mathbf{k}\end{equation}\begin{equation}\|\mathbf{e}_\theta \times \mathbf{e}_r\|^2 = h_\theta^2 + r^2 (h_r^2 + 1)\end{equation}\begin{equation}\text{Àrea (S)} = \iint_D \sqrt{h_\theta^2(r, \theta) + r^2 [1 + h_r^2(r, \theta)]} \, d\theta \, dr\end{equation}