Notation and Conventions

We choose the notation set forth in [WH88]. Their chapter has been the foundation of many research papers, is used by geophysicists world-wide, and it is clean and unambiguous.

Vectors

  • vectors and vectorial operators are bold:
    • e.g., \(\mathbf{v}\), \(\boldsymbol{\nabla\cdot}\)
  • tensors are bold and underlined:
    • e.g., \(\mathbf{\underline{v}}\), \(\boldsymbol{\underline{\sigma}}\)
  • time domain variable are lower case:
    • e.g., \(\mathbf{e}\), \(\mathbf{j}\), \(\mathbf{h}\), \(\mathbf{b}\)
  • frequency domain variables are upper case:
    • e.g., \(\mathbf{E}\), \(\mathbf{J}\), \(\mathbf{H}\), \(\mathbf{B}\)

Integrals

  • Integrating a scalar function over a volume:
    \[\int_V f ~dv\]
    or over a closed volume:
    \[\oint_V f ~dv\]
  • Integrating a vector function over a surface:
    \[\int_S \mathbf{f} \cdot \mathbf{da} = \int_S \mathbf{f} \cdot \mathbf{\hat{n}} ~da\]
    or over a closed surface:
    \[\oint_S \mathbf{f} \cdot \mathbf{da} = \oint_S \mathbf{f} \cdot \mathbf{\hat{n}} ~da\]
  • Integrating a vector function over a curve:
    \[\int_C \mathbf{f} \cdot \mathbf{dl} = \int_C \mathbf{f} \cdot \mathbf{\hat{n}} ~dl\]
    or over a closed curve:
    \[\oint_C \mathbf{f} \cdot \mathbf{dl} = \oint_C \mathbf{f} \cdot \mathbf{\hat{n}} ~dl\]

Fourier Transform Convention

We also adopt their choice of sign in the Fourier Transform: \(e^{i\omega t}\) time dependence.

(1)\[F(\omega) = \int_{-\infty}^{\infty} f(t)e^{-i\omega t} dt\]
(2)\[f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega\]