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12.2: Vector Operators

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  • This section contains a summary of vector operators expressed in each of the three major coordinate systems:

    • Cartesian (\(x\),\(y\),\(z\))
    • cylindrical (\(\rho\),\(\phi\),\(z\))
    • spherical (\(r\),\(\theta\),\(\phi\))

    Associated basis vectors are identified using a caret (\(\hat{~}\)) over the symbol. The vector operand \({\bf A}\) is expressed in terms of components in the basis directions as follows:

    • Cartesian: \({\bf A} = \hat{\bf x}A_x + \hat{\bf y}A_y + \hat{\bf z}A_z\)
    • cylindrical: \({\bf A} = \hat{\bf \rho}A_{\rho} + \hat{\bf \phi}A_{\phi} + \hat{\bf z}A_z\)
    • spherical: \({\bf A} = \hat{\bf r}A_r + \hat{\bf \theta}A_{\theta} + \hat{\bf \phi}A_{\phi}\)


    Gradient in Cartesian coordinates:

    \begin{align} \nabla f &= \hat{\bf x}\frac{\partial f}{\partial x} + \hat{\bf y}\frac{\partial f}{\partial y} + \hat{\bf z}\frac{\partial f}{\partial z} & \end{align}

    Gradient in cylindrical coordinates:

    \begin{align} \nabla f &= \hat{\bf \rho}\frac{\partial f}{\partial \rho} +\hat{\bf \phi}\frac{1}{\rho}\frac{\partial f}{\partial \phi} + \hat{\bf z}\frac{\partial f}{\partial z} &\end{align}

    Gradient in spherical coordinates:

    \begin{align} \nabla f &= \hat{\bf r}\frac{\partial f}{\partial r} +\hat{\bf \theta}\frac{1}{r}\frac{\partial f}{\partial \theta} +\hat{\bf \phi}\frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi} &\end{align}


    Divergence in Cartesian coordinates:

    \begin{align} \nabla \cdot {\bf A} &= \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} &\end{align}

    Divergence in cylindrical coordinates:

    \begin{align} \nabla \cdot {\bf A} &= \frac{1}{\rho}\frac{\partial}{\partial \rho}\left(\rho A_{\rho}\right) +\frac{1}{\rho}\frac{\partial A_{\phi}}{\partial \phi} + \frac{\partial A_z}{\partial z} &\end{align}

    Divergence in spherical coordinates:

    \begin{align} \nabla \cdot {\bf A} &= ~~\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2 A_r\right) & \nonumber \\ &~~ +\frac{1}{r\sin\theta}\frac{\partial}{\partial \theta}\left(A_{\theta}\sin\theta\right)& \nonumber \\ &~~ +\frac{1}{r\sin\theta}\frac{\partial A_{\phi}}{\partial \phi} & \end{align}


    Curl in Cartesian coordinates:

    \begin{align} \nabla \times {\bf A} &= ~~\hat{\bf x}\left( \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z} \right) & \nonumber \\ &~~ +\hat{\bf y}\left( \frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x} \right) & \nonumber \\ &~~ +\hat{\bf z}\left( \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} \right) & \label{m0139_eCurlCart}\end{align}

    Curl in cylindrical coordinates:

    \begin{align} \nabla \times {\bf A} &= ~~\hat{\bf \rho}\left( \frac{1}{\rho}\frac{\partial A_z}{\partial \phi} - \frac{\partial A_{\phi}}{\partial z} \right) & \nonumber \\ &~~ +\hat{\bf \phi}\left( \frac{\partial A_{\rho}}{\partial z} - \frac{\partial A_z}{\partial \rho} \right) & \nonumber \\ &~~ +\hat{\bf z}\frac{1}{\rho}\left[ \frac{\partial}{\partial \rho}\left(\rho A_{\phi}\right) - \frac{\partial A_{\rho}}{\partial \phi} \right] &\end{align}

    Curl in spherical coordinates:

    \begin{align} \nabla \times {\bf A} &= ~~\hat{\bf r}\frac{1}{r\sin\theta} \left[ \frac{\partial}{\partial \theta}\left(A_{\phi}\sin\theta\right) - \frac{\partial A_{\theta}}{\partial \phi} \right] & \nonumber \\ &~~ +\hat{\bf \theta}\frac{1}{r}\left[ \frac{1}{\sin\theta}\frac{\partial A_r}{\partial \phi} - \frac{\partial}{\partial r}\left(rA_{\phi}\right) \right] & \nonumber \\ &~~ +\hat{\bf \phi}\frac{1}{r}\left[ \frac{\partial}{\partial r}\left(r A_{\theta}\right) - \frac{\partial A_r}{\partial \theta} \right] \label{m0139_eCurlSph} &\end{align}


    Laplacian in Cartesian coordinates:

    \begin{align} \nabla^2 f &= \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} &\end{align}

    Laplacian in cylindrical coordinates:

    \begin{align} \nabla^2 f &= \frac{1}{\rho}\frac{\partial }{\partial \rho}\left(\rho\frac{\partial f}{\partial \rho} \right) + \frac{1}{\rho^2}\frac{\partial^2 f}{\partial \phi^2} + \frac{\partial^2 f}{\partial z^2} &\end{align}

    Laplacian in spherical coordinates:

    \begin{align} \nabla^2 f &= ~~\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial f}{\partial r} \right) & \nonumber \\ &~~ +\frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\frac{\partial f}{\partial \theta}\sin\theta\right) & \nonumber \\ &~~ +\frac{1}{r^2\sin^2\theta}\frac{\partial^2 f}{\partial \phi^2} &\end{align}