Math Refresher: Calculus

These appendix pages collect the mathematical tools the rest of the site uses. Each tool gets a definition and one example, plus a pointer to where it first appears. A reader meeting calculus for the first time can follow from these pages; a reader who has seen it can skim.

1. Derivatives

A derivative measures how fast a quantity changes. For a function $q(t)$, the derivative

$$q^{\prime }(t)=\frac{dq}{dt}=\underset{h\to 0}{\lim }\frac{q(t+h)-q(t)}{h}$$

is the slope of its graph, or the rate at which $q$ changes per unit change in $t$. A dot abbreviates a time derivative, $\dot{q}=dq/dt$.

Example. For $q(t)=t^2$, the change over a small interval is $(t+h{)}^2-t^2=2th+h^2$. Dividing by $h$ and letting $h\to 0$ leaves $2t$, so $\dot{q}=2t$. This is why Classical Mechanics describes a particle's velocity as $\dot{q}$ and its acceleration as $\ddot{q}$.

A derivative can be negative or zero. Zero means the quantity is momentarily not changing; this is the condition that marks a maximum or minimum of a function.

2. Partial derivatives

A quantity often depends on several variables at once, such as the Lagrangian $L(q,\dot{q},t)$. A partial derivative differentiates with respect to one variable while holding the others fixed:

$$\frac{\partial L}{\partial \dot{q}}={\frac{dL}{d\dot{q}}|}_{q,t\text{fixed}}.$$

We use the curly $\partial$ to distinguish this from a full derivative $\frac{d}{dt}$, which tracks how a quantity changes when everything that depends on time changes.

Example. For $L=\frac{1}{2}m{\dot{q}}^2-V(q)$, treat $\dot{q}$ and $q$ as independent variables. Then

$$\frac{\partial L}{\partial \dot{q}}=m\dot{q},\frac{\partial L}{\partial q}=-V^{\prime }(q).$$

Both derivatives appear in the Euler–Lagrange equation on Classical Mechanics:

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right)-\frac{\partial L}{\partial q}=0.$$

The same partial-derivative operation acts on a field $\varphi (t,\mathbf{x})$ on the Action page, where the independent variables are $\varphi$ and each derivative ${\partial }_{\mu }\varphi$.

3. The chain rule

When one variable depends on another, their rates of change compose. For $f(q(t))$,

$$\frac{df}{dt}=\frac{df}{dq}\frac{dq}{dt}.$$

With several variables, the full time derivative collects one term per path of dependence:

$$\frac{df}{dt}=\frac{\partial f}{\partial q}\dot{q}+\frac{\partial f}{\partial t}.$$

Example. The Lorentz time transformation $t^{\prime }=\gamma (t-vx/c^2)$ mixes $t$ and $x$, so time derivatives transform with both terms of the chain rule. The same rule converts between a frame's time derivative and the derivative along a particle's own clock when the Lorentz factor appears.

4. Integrals and integration by parts

An integral adds a quantity over a range. The definite integral ${\int }_a^{b}f(t)dt$ accumulates $f$ between $t=a$ and $t=b$; the indefinite integral is the reverse of differentiating.

The action is a definite integral, $S=\int Ldt$. To find which path makes it stationary, we move a derivative off the variation using integration by parts. For two functions $u(t)$ and $v(t)$,

$${\int }_a^{b}u\frac{dv}{dt}dt=[uv{]}_a^b-{\int }_a^{b}v\frac{du}{dt}dt,$$

where $[uv{]}_a^b=u(b)v(b)-u(a)v(a)$ is the boundary term.

Example. In the variation of the action, the velocity term is $\int \frac{\partial L}{\partial \dot{q}}\frac{d}{dt}(\delta q)dt$. Taking $u=\frac{\partial L}{\partial \dot{q}}$ and $v=\delta q$ moves the derivative onto $u$. The boundary term vanishes because the variation $\delta q$ is chosen to be zero at the endpoints. What remains is $\int \left(\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}\right)\delta qdt$, which produces the Euler–Lagrange equation. The same step runs over every spacetime direction when deriving the field equation.

5. Taylor expansion

A smooth function near a point agrees with a polynomial built from its derivatives there:

$$f(x)\approx f(x_0)+f^{\prime }(x_0)(x-x_0)+\frac{1}{2}{f}^{\prime \prime }(x_0)(x-x_0{)}^2+\cdots$$

Each added term fixes one more derivative at $x_0$.

Example. Near a minimum of a potential, $V^{\prime }(x_0)=0$, so the linear term drops out and

$$V(x)\approx V(x_0)+\frac{1}{2}{V}^{\prime \prime }(x_0)(x-x_0{)}^2.$$

The harmonic oscillator is exactly this quadratic approximation. A second use appears in Special Relativity: the Lorentz factor obeys $\gamma \approx 1+v^2/(2c^2)$ at small speed, which recovers the non-relativistic kinetic energy.

6. Vector calculus

Fields vary over space, so we need derivatives that act on all three directions at once. Write $\mathbf{v}=(v_x,v_y,v_z)$ and $\nabla =({\partial }_x,{\partial }_y,{\partial }_z)$.

• Gradient, $\nabla f=({\partial }_{x}f,{\partial }_{y}f,{\partial }_{z}f)$: the vector pointing in the direction $f$ increases most steeply.
• Divergence, $\nabla \cdot \mathbf{v}={\partial }_x{v}_x+{\partial }_y{v}_y+{\partial }_z{v}_z$: the net outward flow of $\mathbf{v}$ per unit volume.
• Curl, $\nabla \times \mathbf{v}$: a vector measuring circulation of $\mathbf{v}$ around each axis.
• Laplacian, ${\nabla }^{2}f=\nabla \cdot (\nabla f)={\partial }_x^{2}f+{\partial }_y^{2}f+{\partial }_z^{2}f$: the divergence of the gradient.

The multidimensional analogue of the integration by parts in §4 is the divergence theorem:

$${\int }_V\nabla \cdot \mathbf{v}{d}^{3}x={\oint }_{\partial V}\mathbf{v}\cdot \mathbf{n}dA.$$

The integral of a divergence over a volume equals the flux of the vector through the volume's boundary. It turns a continuity equation into a conservation statement: a change in a quantity inside a region equals the net flow through the boundary, so a flow that vanishes at infinity leaves the total unchanged, as in the probability current argument.

These appear in the Maxwell equations: $\nabla \cdot \mathbf{E}=\rho$, $\nabla \times \mathbf{E}=-{\partial }_t\mathbf{B}$, and so on. The relativistic wave operator combines a time derivative with the Laplacian,

$$\square ={\partial }_t^2-{\nabla }^2,$$

called the d'Alembertian, which appears in the Klein–Gordon equation.

Next: Complex Numbers