Appendix A: Mathematical Preliminaries

This appendix is internal to the Classical Physics book and is not discoverable outside of it.

Gradient, Divergence, Curl

$$\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$$ $$\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$$ $$\nabla \times \mathbf{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \partial_x & \partial_y & \partial_z \\ F_x & F_y & F_z \end{vmatrix}$$

First-Order ODEs

Separable form: $\frac{dy}{dx} = f(x)g(y)$ integrates to $\int \frac{dy}{g(y)} = \int f(x)\,dx$.