Differential and Integral Notation

We will generally employ Leibniz's notation for differentiation and anti-differentiation. The derivative of a function of one variable, e.g. $f(x) = f$, where $x$ is the independent variable, is written:

$$ \frac{\mathrm{d}f}{\mathrm{d}x} \text{ or } \frac{\mathrm{d}}{\mathrm{d}x}f $$

And higher-order derivatives are written as:

$$ \frac{\mathrm{d}^2f}{\mathrm{d}x^2}, \frac{\mathrm{d}^3f}{\mathrm{d}x^3}, ..., \frac{\mathrm{d}^nf}{\mathrm{d}x^n} $$

You will encounter a partial differential equation during this course that describes time-dependent diffusion in one spatial dimension (Fick's second law) - imagine a gas diffusing into a tube. You will not be required to solve differential equations, but you will have to interpret them and test (provided) solutions to the equations.

Partial differential equations with multiple variables use the same notation as above, but utilize the $\partial$ symbol. Here we define the $g(x,t) = g$, where $x$ and $t$ are independent variables:

\begin{equation} \frac{\partial g}{\delta x} \text{ or } \frac{\partial }{\delta x}g \end{equation}

$$ \frac{\partial^2 g}{\partial x^2}, \frac{\partial^3g}{\partial x^3}, ..., \frac{\partial^n g}{\partial x^n} $$

Antidifferentiation will be denoted using the integral symbol, e.g. for the definite integration of $x^2$ from $a$ to $b$:

$$ \int_{a}^{b} x^2 \mathrm{d}x $$

After integration, evaluation of this definite integral is written as:

$$ \frac{x^3}{3} \Big|_a^b $$

Below, we use Lagrange shorthand to denote derivatives, i.e. $\frac{\mathrm{d}}{\mathrm{d}x} = f^{\prime}(x)$.