Conventions =========== Coordinate systems ------------------ Cartesian ^^^^^^^^^ We use a right-handed coordinate system :math:(x, y, z) with z-positive up as shown in the Figure below. .. image:: ../images/coordinates-02.png :width: 40% :align: center Cylindrical ^^^^^^^^^^^ We again work with z-positive up and use :math:\theta to denote the azimuthal angle, thus the coordinate system is defined as :math:(r, \theta, z). .. image:: ../images/coordinates-03.png :width: 40% :align: center Spherical ^^^^^^^^^ We use :math:r for the radial direction, :math:\theta for the azimuthal direction, and :math:\phi for the polar direction as shown in the figure below. .. image:: ../images/coordinates-04.png :width: 40% :align: center Fourier Transform ----------------- For analysis and solutions in the frequency domain we use the :math:e^{i \omega t} Fourier transform convention. Thus, we define our Fourier Transform pair as .. math :: F(\omega) = \int_{-\infty}^{\infty} f(t) e^{- i \omega t} dt \\ f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega) e^{i \omega t} d \omega where :math:\omega is angular frequency, :math:t is time, :math:F(\omega) is the function defined in the frequency domain and :math:f(t) is the function defined in the time domain.