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Engineering LibreTexts

20.2: I.2 Spherical coordinates

  • Page ID
    18119
  • Let \(\psi\) be a scalar and \(\vec{u}\) be a vector expressed as a linear combination of the cylindrical basis vectors: \(\vec{u} = u_r\hat{e}^{(r)} +u_\theta \hat{e}^{(\theta)} +u_\phi \hat{e}^{(\phi)}\).

    Gradient of a scalar

    \[\vec{\nabla} \psi=\hat{e}^{(r)} \frac{\partial \psi}{\partial r}+\hat{e}^{(\theta)} \frac{1}{r} \frac{\partial \psi}{\partial \theta}+\hat{e}^{(\phi)} \frac{1}{r \sin \theta} \frac{\partial \psi}{\partial \phi}\label{eqn:1}\]

    Divergence of a vector

    \[\vec{\nabla} \cdot \vec{u}=\frac{1}{r^{2}} \frac{\partial\left(r^{2} u_{r}\right)}{\partial r}+\frac{1}{r \sin \theta} \frac{\partial\left(u_{\theta} \sin \theta\right)}{\partial \theta}+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial u_{\phi}}{\partial \phi}\label{eqn:2}\]

    Curl of a vector

    \[\vec{\nabla} \times \vec{u}=\hat{e}^{(r)} \frac{1}{r \sin \theta}\left[\frac{\partial\left(u_{\phi} \sin \theta\right)}{\partial \theta}-\frac{\partial u_{\theta}}{\partial \phi}\right]+\hat{e}^{(\theta)} \frac{1}{r}\left[\frac{1}{\sin \theta} \frac{\partial u_{r}}{\partial \phi}-\frac{\partial\left(r u_{\phi}\right)}{\partial r}\right]+\hat{e}^{(\phi)} \frac{1}{r}\left[\frac{1}{r} \frac{\partial\left(r u_{\theta}\right)}{\partial r}-\frac{\partial u_{r}}{\partial \theta}\right]\label{eqn:3}\]

    Laplacian of a scalar

    \[\nabla^{2} \psi=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial \psi}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial \psi}{\partial \theta}\right)+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial^{2} \psi}{\partial \phi^{2}}\label{eqn:4}\]

    Laplacian of a vector

    \[\begin{equation}\begin{aligned}
    \nabla^{2} \vec{u} &=\hat{e}^{(r)}\left[\nabla^{2} u_{r}-\frac{2 u_{r}}{r^{2}}-\frac{2}{r^{2} \sin \theta} \frac{\partial\left(u_{\theta} \sin \theta\right)}{\partial \theta}-\frac{2}{r^{2} \sin \theta} \frac{\partial u_{\phi}}{\partial \phi}\right] \\
    &+\hat{e}^{(\theta)}\left[\nabla^{2} u_{\theta}+\frac{2}{r^{2}} \frac{\partial u_{r}}{\partial \theta}-\frac{u_{\theta}}{r^{2} \sin ^{2} \theta}-\frac{2}{r^{2}} \frac{\cos \theta}{\sin ^{2} \theta} \frac{\partial u_{\phi}}{\partial \phi}\right] \\
    &+\hat{e}^{(\phi)}\left[\nabla^{2} u_{\phi}+\frac{2}{r^{2} \sin ^{2} \theta} \frac{\partial u_{r}}{\partial \phi}+\frac{2}{r^{2}} \frac{\cos \theta}{\sin ^{2} \theta} \frac{\partial u_{\theta}}{\partial \phi}-\frac{u_{\theta}}{r^{2} \sin ^{2} \theta}\right]
    \end{aligned}\label{eqn:5}\end{equation}\]

    Material derivative

    \[\frac{D}{D t}=\frac{\partial}{\partial t}+u_{r} \frac{\partial}{\partial r}+\frac{u_{\theta}}{r} \frac{\partial}{\partial \theta}+\frac{u_{\phi}}{r \sin \theta} \frac{\partial}{\partial \phi}\label{eqn:6}\]

    Centrifugal acceleration

    \[\vec{a}_{C}=\hat{e}^{(r)} \frac{u_{\theta}^{2}+u_{\phi}^{2}}{r}+\hat{e}^{(\theta)}\left(-\frac{u_{r} u_{\theta}}{r}+\frac{u_{\phi}^{2}}{r \tan \theta}\right)+\hat{e}^{(\phi)}\left(-\frac{u_{r} u_{\phi}}{r}-\frac{u_{\theta} u_{\phi}}{r \tan \theta}\right).\]