Maths

A reminder to myself of some useful mathematical tools.

Fourier Transform

The Fourier transform allows us to move between the temporal domain and the frequency domain:

$$F(\omega) = \int_{-\infty}^{\infty} f(t)e^{-i\omega t}\,dt$$

The problem is that the Fourier transform is only defined for functions that are sufficiently integrable over $[-\infty,\infty]$.

For example, consider:

$$x(t)=e^t$$

Its Fourier transform would involve:

$$\int_{-\infty}^{\infty} e^t e^{-i\omega t}\,dt = \int_{-\infty}^{\infty} e^{(-i\omega+1)t}\,dt$$

However, this integral does not converge because:

$$\operatorname{Re}(1-i\omega)=1>0$$

Laplace Transform

The Laplace transform generalizes the Fourier transform by introducing a complex variable $s$:

$$F(s) = \int_{-\infty}^{\infty} f(t)e^{-st}\,dt$$

with:

$$s=\sigma+i\omega$$

This allows us to integrate a broader class of functions, including functions bounded by exponentials:

$$|f(t)| \leq Ae^{\alpha t}, \qquad (A,\alpha)\in\mathbb{R}$$

provided that:

$$\alpha < \sigma$$

Convolution

The convolution of two functions $f$ and $g$ is defined as:

$$(f*g)(t) = \int_{-\infty}^{\infty} f(\tau)g(t-\tau)\,d\tau$$

An important property of the Laplace transform is that convolution in the time domain becomes multiplication in the Laplace domain:

$$L(f*g)(s)=F(s)G(s)$$

Vector Calculus

Nabla operator

The nabla operator is defined as:

$$\vec{\nabla} = \begin{pmatrix} \frac{\partial}{\partial x}\\ \frac{\partial}{\partial y}\\ \frac{\partial}{\partial z} \end{pmatrix}$$

Divergence

The divergence applies the nabla operator to a vector field:

$$\vec{\nabla}\cdot\vec{F} = \begin{pmatrix} \frac{\partial}{\partial x}\\ \frac{\partial}{\partial y}\\ \frac{\partial}{\partial z} \end{pmatrix} \cdot \begin{pmatrix} F_x\\ F_y\\ F_z \end{pmatrix}$$

which gives:

$$\vec{\nabla}\cdot\vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$$

The divergence therefore maps a vector field to a scalar field:

$$E \rightarrow C$$

Gradient

The nabla operator can also be applied to a scalar field $f$:

$$\vec{\nabla}f = \begin{pmatrix} \frac{\partial}{\partial x}\\ \frac{\partial}{\partial y}\\ \frac{\partial}{\partial z} \end{pmatrix} f$$

giving:

$$\vec{\nabla}f = \begin{pmatrix} \frac{\partial f}{\partial x}\\ \frac{\partial f}{\partial y}\\ \frac{\partial f}{\partial z} \end{pmatrix}$$

The gradient therefore maps a scalar field to a vector field:

$$C \rightarrow E$$

Curl

The curl of a vector field $\vec{F}$ is:

$$\vec{\nabla}\times\vec{F} = \begin{pmatrix} \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \\ \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \\ \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \end{pmatrix}$$

The curl describes the local rotational behavior of a vector field.

Green–Ostrogradsky Theorem

The Green–Ostrogradsky theorem, also known as the divergence theorem, relates the divergence of a vector field over a volume to its flux through the boundary surface:

$$\int_V \vec{\nabla}\cdot\vec{F}\,dV = \int_S \vec{F}\cdot\vec{n}\,dS$$

where $\vec{n}$ is the outward-pointing normal vector to the surface $S$.