Vector Analysis

Assumed that person reading this already possess a decent knowledge about this topic. Only special things are noted down.

  1. Cylindrical coordinates \((x,y,z) \rightarrow (\rho, \phi, z)\)
  2. Spherical coordinates \((x,y,z) \rightarrow (r, \theta, \phi)\)
  3. Differential length (\(\vec{dl}\)), surface (\(\vec{ds}\)), volume (\(dv\))
  4. Del operator \(\nabla\)
  5. Divergence
  6. Curl
  7. Laplacian of a Scalar

Scalar triple product:

\[\vec{A} \cdot (\vec{B} \times \vec{C}) = \vec{B} \cdot (\vec{C} \times \vec{A}) = \vec{C} \cdot (\vec{A} \times \vec{B}) = \begin{vmatrix} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \\ \end{vmatrix}\]

Vector triple product: (bac-cab rule)

\[\vec{A} \times (\vec{B} \times \vec{C}) = \vec{B}(\vec{A} \cdot \vec{C}) - \vec{C}(\vec{A} \cdot \vec{B})\]

Cylindrical coordinates \((x,y,z) \rightarrow (\rho, \phi, z)\)

  • Ranges: \(0 \leq\rho\leq\infty, \quad 0\leq\phi\leq 2\pi, \quad -\infty\leq z \leq\infty\)
  • \[\hat{a_\rho} \times \hat{a_{\phi}} = \hat{a_z}\]
  • \[\rho = \sqrt{x^2 + y^2}, \quad \phi = \tan^{-1}\frac{y}{x}, \quad z = z\]
  • \[x = \rho \cos \phi, \quad y = \rho \sin \phi, \quad z=z\]
  • Transformation: \((A_x\hat{a_x} + A_y\hat{a_y} + A_z\hat{a_z}) \longleftrightarrow (A_\rho\hat{a_\rho} + A_\phi\hat{a_\phi} + A_z\hat{a_z})\)
\[\begin{bmatrix} A_\rho \\ A_\phi \\ A_z \end{bmatrix} = \begin{bmatrix} \cos\phi & \sin\phi & 0 \\ -\sin\phi & \cos\phi & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} A_x \\ A_y \\ A_z \end{bmatrix}\] \[\begin{bmatrix} A_x \\ A_y \\ A_z \end{bmatrix} = \begin{bmatrix} \cos\phi & -\sin\phi & 0 \\ \sin\phi & \cos\phi & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} A_\rho \\ A_\phi \\ A_z \end{bmatrix}\]

Spherical coordinates \((x,y,z) \rightarrow (r, \theta, \phi)\)

  • Ranges: \(0 \leq r \leq\infty, \quad 0\leq\theta\leq \pi, \quad 0\leq \phi \leq 2\pi\)
  • \[\hat{a_r} \times \hat{a_\theta} = \hat{a_\phi}\]
  • \[r = \sqrt{x^2 + y^2 + z^2}, \quad \theta = \tan^{-1}\frac{\sqrt{x^2+y^2}}{z}, \quad \phi = \tan^{-1}\frac{y}{x}\]
  • \[x = r\sin\theta\cos\phi, \quad y = r\sin\theta\sin\phi, \quad z=r\cos\theta\]
  • Transformation: \((A_x\hat{a_x} + A_y\hat{a_y} + A_z\hat{a_z}) \longleftrightarrow (A_r\hat{a_r} + A_\theta\hat{a_\theta} + A_\phi\hat{a_\phi})\)
\[\begin{bmatrix} A_r \\ A_\theta \\ A_\phi \end{bmatrix} = \begin{bmatrix} \sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta \\ \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta \\ -\sin\phi & \cos\phi & 0 \\ \end{bmatrix} \begin{bmatrix} A_x \\ A_y \\ A_z \end{bmatrix}\] \[\begin{bmatrix} A_x \\ A_y \\ A_z \end{bmatrix} = \begin{bmatrix} \sin\theta\cos\phi & \cos\theta\cos\phi & -\sin\phi \\ \sin\theta\cos\phi & \cos\theta\sin\phi & \cos\phi \\ \cos\theta & -\sin\theta & 0 \\ \end{bmatrix} \begin{bmatrix} A_r \\ A_\theta \\ A_\phi \end{bmatrix}\]

Differential length (\(\vec{dl}\)), surface (\(\vec{ds}\)), volume (\(dv\))

  • Cartesian:
    \(\begin{align*} \vec{dl} &= dx \space \vec{a_x} + dy \space \vec{a_y} + dx \space \vec{a_x} \\ \vec{ds} &= dy \space dz \space \vec{a_x} + dx \space dz \space \vec{a_y} + dx \space dy \space \vec{a_x} \\ dv &= dx \space dy \space dz \end{align*}\)

  • Cylindrical:
    \(\begin{align*} \vec{dl} &= d\rho \space \vec{a_\rho} + \rho d\phi \space \vec{a_\phi} + dx \space \vec{a_x} \\ \vec{ds} &= \rho d\phi \space dz \space \vec{a_\phi} + d\rho \space dz \space \vec{a_\phi} + \rho d\rho \space d\phi \space \vec{a_z} \\ dv &= \rho d\rho \space d\phi \space dz \end{align*}\)

  • Spherical:
    \(\begin{align*} \vec{dl} &= dr \space \vec{a_r} + r d\phi \space \vec{a_\theta} + r \sin \theta d\phi \space \vec{a_\phi} \\ \vec{ds} &= r^2 \sin\theta \space d\theta \space d\phi \space \vec{a_r} + r\sin\theta \space dr \space d\phi \space \vec{a_\theta} + r dr \space d\theta \space \vec{a_\phi} \\ dv &= r^2 \sin\theta \space dr \space d\theta \space d\phi \end{align*}\)

Del operator \(\nabla\)

Cartesian: \(\nabla = \frac{\partial}{\partial x}\hat{a_x} + \frac{\partial}{\partial y}\hat{a_y} + \frac{\partial}{\partial z}\hat{a_z}\)

Cylindrical: \(\nabla = \frac{\partial}{\partial \rho}\hat{a_\rho} + \frac{1}{\rho}\frac{\partial}{\partial \phi}\hat{a_\phi} + \frac{\partial}{\partial z}\hat{a_z}\)

Spherical: \(\nabla = \frac{\partial}{\partial r}\hat{a_r} + \frac{1}{r}\frac{\partial}{\partial \theta}\hat{a_\theta} + \frac{1}{r\sin\theta}\frac{\partial}{\partial \phi}\hat{a_\phi}\)

  • gradient: \(\nabla V\)
  • divergence: \(\nabla \cdot \vec{A}\)
  • curl: \(\nabla \times \vec{A}\)
  • Laplacian: \(\nabla^2V\)
  • \[\nabla \cdot (\nabla \times \vec{A}) = 0\]
  • \[\nabla \times (\nabla f) = 0\]
  • \[\nabla \times (\nabla \times \vec{A}) = \nabla(\nabla \cdot \vec{A}) - \nabla^2 \vec{A}\]
  • \[\nabla(UV) = V\nabla U + U \nabla V\]

Divergence

How much a field diverges from a point

Cartesian: \(\nabla \cdot \vec{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}\)

Cylindrical: \(\nabla \cdot \vec{A}= \frac{1}{\rho}\frac{\partial (\rho A_\rho)}{\partial \rho} + \frac{1}{\rho}\frac{\partial A_\phi}{\partial \phi} + \frac{\partial A_z}{\partial z}\)

Spherical: \(\nabla \cdot \vec{A}= \frac{1}{r^2}\frac{\partial(r^2 A_r)}{\partial r} + \frac{1}{r\sin \theta}\frac{\partial(A_\theta \sin\theta)}{\partial \theta} + \frac{1}{r\sin\theta}\frac{\partial A_\phi}{\partial \phi}\)

Divergence Theorem: \(\oint_S \vec{A} \cdot d\vec{S} = \int_V \nabla \cdot \vec{A} dv\)

Curl

whether there is a rotation associated with a vector field

Cartesian: \(\nabla \times \vec{A} = \begin{vmatrix} \hat{a_x} & \hat{a_y} & \hat{a_z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_x & A_y & A_z \end{vmatrix}\)

Cylindrical: \(\nabla \times \vec{A} = \frac{1}{\rho}\begin{vmatrix} \hat{a_\rho} & \rho \hat{a_\phi} & \hat{a_z} \\ \frac{\partial}{\partial \rho} & \frac{\partial}{\partial \phi} & \frac{\partial}{\partial z} \\ A_\rho & \rho A_\phi & A_z \end{vmatrix}\)

Spherical: \(\nabla \times \vec{A} = \frac{1}{r^2 \sin\theta}\begin{vmatrix} \hat{a_r} & r\hat{a_\theta} & r \sin\theta \hat{a_\phi} \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial \theta} & \frac{\partial}{\partial \phi} \\ A_r & rA_\theta & r \sin\theta A_\phi \end{vmatrix}\)

Stokes’ Theorem: \(\oint_L \vec{A} \cdot d\vec{l} = \int_S (\nabla \times \vec{A}) \cdot d\vec{S}\)

Laplacian of a Scalar

Cartesian: \(\nabla^2 V = \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2}\)

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

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