Partial derivatives and vector calculus

Gradient, divergence, curl, the Laplacian.

Once a quantity depends on more than one variable — pressure $p(x, y, z, t)$, velocity $\mathbf{v}(x, y, z, t)$ — we need partial derivatives and vector operators. This chapter collects the ones the Sound book uses constantly, organised across three lessons.

• 2.1 Partial derivatives and the gradient — partial derivatives, the total differential, the gradient $\nabla\phi$, and the directional derivative that ties the gradient to a geometric “steepest ascent” interpretation.
• 2.2 Divergence and curl — divergence (outflow per unit volume), curl (local rotation), the two identities $\nabla \times \nabla\phi = 0$ and $\nabla \cdot \nabla\times\mathbf{A} = 0$, plus a sketch of the late-19th-century notation war that gave us these operators in their modern form.
• 2.3 The Laplacian and harmonic functions — the second-order Laplacian $\nabla^2 \phi$, harmonic functions, and the canonical role of $\nabla^2$ in every linear PDE the bookshelf cares about.