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