Vector Fields and Integral Theorems

A vector field $\mathbf{F}:\mathbb{R}^n\to\mathbb{R}^n$ assigns to each point $\mathbf{x}$ a vector $\mathbf{F}(\mathbf{x})$. Unlike a scalar field (one number per point, like temperature), here each point carries an arrow: a direction and a magnitude.

Physical examples. The velocity field of a fluid (at each point the velocity of the particle there), the gravitational field (force per unit mass), the electric field (force per unit charge). Drawing them helps intuition: draw an arrow at each point, or the flow lines (curves tangent to the field everywhere, like iron filings around a magnet).

Flow lines. They are the trajectories of a particle carried by the field: they solve the system $\mathbf{r}'(t)=\mathbf{F}(\mathbf{r}(t))$ — a direct link to differential equations. Where the lines crowd together, the field is strong.

Two questions guide the study: does the field "flow out" of a point? (divergence) and does the field "rotate" around a point? (curl). These are the two fundamental derivatives of the next section.

For $\mathbf{F}=(P,Q,R)$ we define two differential operators, built formally with the "nabla" symbol $\nabla=(\partial_x,\partial_y,\partial_z)$.

Divergence (scalar):

It measures the net outward flux of the field per unit volume, in an infinitesimal neighbourhood of the point. Positive divergence = source (the field "gushes out", as from a positive charge); negative = sink (the field is "drained in"); zero = solenoidal field (whatever flows in flows out — like the magnetic field).

Curl (vector):

It measures the tendency to rotate around a point. If you place a tiny paddle wheel in the field, the curl tells whether and how fast it spins, and about which axis. Zero curl = irrotational field. It is conveniently computed as a formal determinant with $\nabla$ and the unit vectors $\hat i,\hat j,\hat k$.

A field $\mathbf{F}$ is conservative if it derives from a scalar potential $U$: $\mathbf{F}=\nabla U$. For these fields the work depends only on the endpoints (seen in the Curves chapter), and the circulation around any closed path is zero.

The curl test. On a simply connected domain of $\mathbb{R}^3$ (no "holes" the path can encircle):

That is, conservative $\iff$ irrotational. The deep reason: the curl of a gradient is always zero, $\nabla\times(\nabla U)=\mathbf{0}$ (mixed second partials commute). So $\nabla\times\mathbf{F}=\mathbf{0}$ is necessary; simple connectedness makes it sufficient too.

Consequences. $\oint_\gamma\mathbf{F}\cdot d\mathbf{r}=0$ on every closed curve, and $\int_\gamma\mathbf{F}\cdot d\mathbf{r}=U(B)-U(A)$. In physics: gravity and the electrostatic force are conservative (well-defined potential energy); friction is not.

Mind the domain. On domains with holes the test can mislead: the "vortex" field $\mathbf{F}=(-y,x)/(x^2+y^2)$ is irrotational on $\mathbb{R}^2\setminus\{0\}$ but not conservative ($\oint=2\pi$ around the origin).

The flux of a field $\mathbf{F}$ through an oriented surface $\Sigma$ measures "how much field crosses the surface". One projects $\mathbf{F}$ onto the normal $\hat n$ and integrates over the area:

The vector $\mathbf{r}_u\times\mathbf{r}_v$ is normal to the surface (seen in the Vector functions chapter) and already contains the area element: its magnitude is $dS$, its direction the orientation.

Physical meaning. If $\mathbf{F}$ is the velocity of a fluid, $\Phi$ is the flow rate: the volume of fluid crossing $\Sigma$ per unit time. Only the component of $\mathbf{F}$ perpendicular to the surface contributes; flow parallel to it does not "cross".

Orientation. The sign of the flux depends on which direction of the normal is chosen. For closed surfaces one conventionally uses the outward normal. Reversing the orientation flips the flux's sign.

The first of the three integral theorems, in two dimensions. It links a line integral over the boundary of a domain to a double integral over its interior.

For a plane domain $D$ with boundary $\partial D$ oriented positively (counterclockwise, keeping $D$ on the left):

What it says. The circulation of the field along the boundary equals the integral, over the area, of the "$z$-component of the curl" $Q_x-P_y$. If the field is conservative ($Q_x=P_y$) the right-hand side vanishes: we recover $\oint=0$.

A very useful corollary: area as a boundary integral. Choosing suitable $P,Q$:

This is the principle of the planimeter, the instrument measuring a figure's area by tracing only its contour. Green's theorem is the planar case of Stokes' theorem (§7).

It generalises Green's theorem to 3D, linking a flux through a closed surface to the integral of the divergence over the enclosed volume.

For a volume $V$ with boundary $\partial V$ (closed surface) oriented outward:

The idea. The net outward flux through the surface is the sum of all internal "sources" (the divergence). Internal walls between adjacent cells cancel: only the flux through the outer boundary survives. It is the analogue of the fundamental theorem of calculus ("integral of the derivative = boundary values"), in 3D.

In physics. It is Gauss's law of electrostatics: the flux of the electric field through a closed surface is proportional to the enclosed charge ($\nabla\cdot\mathbf{E}=\rho/\varepsilon_0$). It allows computing fields by exploiting symmetry, without integrating directly.

The third integral theorem: it links the circulation of a field along a closed curve to the flux of its curl through any surface bounded by that curve.

For a surface $\Sigma$ with boundary $\partial\Sigma=\gamma$, coherently oriented (right-hand rule):

What it says. The "overall rotation" measured by going around the boundary equals the sum of the micro-rotations (the curl) over the surface. Remarkably, the result does not depend on which surface you choose, as long as it has the same boundary.

It unifies the previous cases. In a plane ($\mathbf{F}$ 2D), Stokes becomes Green. If $\mathbf{F}$ is conservative ($\nabla\times\mathbf{F}=\mathbf{0}$), the right-hand side vanishes and we recover $\oint=0$.

In physics. It is Ampère's law (the circulation of $\mathbf{B}$ along a circuit = enclosed current) and Faraday's law (the induced emf = rate of change of magnetic flux). Stokes is the mathematical bridge of Maxwell's equations in rotational form.

Green, Divergence and Stokes are the same principle in different dimensions: the integral of a "derivative" over a region equals the integral of the field over the region's boundary. It is the fundamental theorem of calculus, generalised (in modern language: Stokes' theorem for differential forms, $\int_\Omega d\omega=\int_{\partial\Omega}\omega$).

Two key identities that simplify everything: the curl of a gradient is zero, $\nabla\times(\nabla U)=\mathbf{0}$ (conservative fields are irrotational); the divergence of a curl is zero, $\nabla\cdot(\nabla\times\mathbf{F})=0$ (curl fields are solenoidal — no sources for the magnetic field).

Maxwell. The four Maxwell equations are exactly these relations applied to $\mathbf{E}$ and $\mathbf{B}$: two with the divergence (electric and magnetic Gauss), two with the curl (Faraday and Ampère-Maxwell). All of classical electromagnetism lives in this chapter.