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เอกสาร 

1. Vector Calculus (Latex) 

Differential Length

Cartesian d\vb{l} = dx\,\vb{\hat{a}_x} + dy\,\vb{\hat{a}_y} + dz\,\vb{\hat{a}_z}

Cylindrical d\vb{l} = d\rho\,\vb{\hat{a}_\rho} + \rho\,d\phi\,\vb{\hat{a}_\phi} + dz\,\vb{\hat{a}_z}

Spherical d\vb{l} = dr\,\vb{\hat{a}_r} + r\,d\theta\,\vb{\hat{a}_\theta} + r\sin\theta\,d\phi\,\vb{\hat{a}_\phi}

Differential Surface

Cartesian d\vb{S} = dy\,dz\,\vb{\hat{a}_x} + dx\,dz\,\vb{\hat{a}_y} + dx\,dy\,\vb{\hat{a}_z}

Cylindrical d\vb{S} = \rho\,d\phi\,dz\,\vb{\hat{a}_\rho} + d\rho\,dz\,\vb{\hat{a}_\phi} + \rho\,d\rho\,d\phi\,\vb{\hat{a}_z}

Spherical d\vb{S} = r^2\sin\theta\,d\theta\,d\phi\,\vb{\hat{a}_r} + r\sin\theta\,dr\,d\phi\,\vb{\hat{a}_\theta} + r\,dr\,d\theta\,\vb{\hat{a}_\phi}

Differential Volume

Cartesian dV = dx\,dy\,dz

Cylindrical dV = \rho\,d\rho\,d\phi\,dz

Spherical dV = r^2\sin\theta\,dr\,d\theta\,d\phi

Gradient

Cartesian \grad{V} = \frac{\partial V}{\partial x}\vb{\hat{a}_x} + \frac{\partial V}{\partial y}\vb{\hat{a}_y} + \frac{\partial V}{\partial z}\vb{\hat{a}_z}

Cylindrical \grad{V} = \frac{\partial V}{\partial \rho}\vb{\hat{a}_\rho} + \frac{1}{\rho}\frac{\partial V}{\partial \phi}\vb{\hat{a}_\phi} + \frac{\partial V}{\partial z}\vb{\hat{a}_z}

Spherical \grad{V} = \frac{\partial V}{\partial r}\vb{\hat{a}_r} + \frac{1}{r}\frac{\partial V}{\partial \theta}\vb{\hat{a}_\theta} + \frac{1}{r\sin\theta}\frac{\partial V}{\partial \phi}\vb{\hat{a}_\phi}

Divergence

Cartesian \div{\vb{D}} = \frac{\partial D_x}{\partial x} + \frac{\partial D_y}{\partial y} + \frac{\partial D_z}{\partial z}

Cylindrical \div{\vb{D}} = \frac{1}{\rho}\frac{\partial (\rho D_\rho)}{\partial \rho} + \frac{1}{\rho}\frac{\partial D_\phi}{\partial \phi} + \frac{\partial D_z}{\partial z}

Spherical \div{\vb{D}} = \frac{1}{r^2}\frac{\partial (r^2 D_r)}{\partial r} + \frac{1}{r\sin\theta}\frac{\partial (D_\theta \sin\theta)}{\partial \theta} + \frac{1}{r\sin\theta}\frac{\partial D_\phi}{\partial \phi}

Laplacian

Cartesian \laplacian{V} = \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2}

Cylindrical \laplacian{V} = \frac{1}{\rho}\frac{\partial}{\partial \rho}\left(\rho\frac{\partial V}{\partial \rho}\right) + \frac{1}{\rho^2}\frac{\partial^2 V}{\partial \phi^2} + \frac{\partial^2 V}{\partial z^2}

Spherical \laplacian{V} = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial V}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial V}{\partial \theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2 V}{\partial \phi^2}

Curl

Cartesian \curl{\vb{H}} = \left(\frac{\partial H_z}{\partial y} - \frac{\partial H_y}{\partial z}\right)\vb{\hat{a}_x} + \left(\frac{\partial H_x}{\partial z} - \frac{\partial H_z}{\partial x}\right)\vb{\hat{a}_y} + \left(\frac{\partial H_y}{\partial x} - \frac{\partial H_x}{\partial y}\right)\vb{\hat{a}_z}

Cylindrical \curl{\vb{H}} = \left(\frac{1}{\rho}\frac{\partial H_z}{\partial \phi} - \frac{\partial H_\phi}{\partial z}\right)\vb{\hat{a}_\rho} + \left(\frac{\partial H_\rho}{\partial z} - \frac{\partial H_z}{\partial \rho}\right)\vb{\hat{a}_\phi} + \frac{1}{\rho}\left(\frac{\partial (\rho H_\phi)}{\partial \rho} - \frac{\partial H_\rho}{\partial \phi}\right)\vb{\hat{a}_z}

Spherical \curl{\vb{H}} = \frac{1}{r\sin\theta}\left[\frac{\partial (H_\phi \sin\theta)}{\partial \theta} - \frac{\partial H_\theta}{\partial \phi}\right]\vb{\hat{a}_r} + \frac{1}{r}\left[\frac{1}{\sin\theta}\frac{\partial H_r}{\partial \phi} - \frac{\partial (r H_\phi)}{\partial r}\right]\vb{\hat{a}_\theta} + \frac{1}{r}\left[\frac{\partial (r H_\theta)}{\partial r} - \frac{\partial H_r}{\partial \theta}\right]\vb{\hat{a}_\phi}