Divergence and Stokes' theorems

The fundamental theorem of calculus says that integrating a derivative over an interval gives the difference of the function's values at the two ends: the inside is summarized by the boundary. This chapter presents its three big multivariable cousins, Green's theorem, the divergence theorem and Stokes' theorem. Each says that adding up a derivative of a field over a region equals adding up the field itself around the region's boundary.

These theorems are not just elegant mathematics. They are how the conservation laws of engineering are written. The equation of continuity in fluid mechanics, Gauss's and Ampère's laws in electromagnetics, and the heat equation all come from them. The chapter starts with the two derivatives involved, divergence and curl, and what they mean physically, then states and applies each theorem.

Divergence: sources and sinks

The divergence of a vector field \(\mathbf{F} = (P, Q, R)\) is the scalar

Physically, divergence measures the net outflow per unit volume at a point. If \(\mathbf{F}\) is the velocity of a fluid, positive divergence means fluid is spreading out from the point (a source, or a gas expanding); negative divergence means it is converging (a sink, or a gas being compressed). Zero divergence means whatever flows into a small box flows out again.

For a liquid, which is very nearly incompressible, conservation of mass requires \(\nabla \cdot \mathbf{v} = 0\) everywhere except at real sources and drains. This is the continuity equation for incompressible flow, the first equation every fluid mechanics course writes down.

Curl: rotation

The curl of \(\mathbf{F}\) is the vector

Curl measures local rotation. Imagine a tiny paddle wheel placed in a flowing fluid: the curl points along the axis about which it spins fastest (by the right-hand rule), and its magnitude is twice the wheel's angular velocity. A field with zero curl everywhere is called irrotational, and on a region without holes it is exactly a conservative (gradient) field, the 3D version of the test \(P_y = Q_x\) from Lesson 4. Indeed, the z-component of the curl, \(Q_x - P_y\), is precisely that 2D test.

Two identities are worth remembering: the curl of a gradient is always zero, \(\nabla \times (\nabla f) = \mathbf{0}\), and the divergence of a curl is always zero, \(\nabla \cdot (\nabla \times \mathbf{F}) = 0\). Both follow from the equality of mixed partial derivatives. A third combination, the divergence of a gradient, gives the Laplacian, \(\nabla^2 f = f_{xx} + f_{yy} + f_{zz}\), which appears in the heat, wave and Laplace equations.

Green's theorem

Green's theorem connects a line integral around a closed curve C in the plane to a double integral over the region D it encloses. With C traversed counterclockwise (the region on the left):

The integrand on the right is the z-component of the curl. So Green's theorem says: the total circulation around the boundary equals the total rotation inside. The picture explains why. Divide the region into tiny cells and add up the circulation around each one. Along every interior edge, the two neighboring cells traverse it in opposite directions, so those contributions cancel, and only the outer boundary survives.

Area from a boundary

Choosing \(P = -y/2\), \(Q = x/2\) makes \(Q_x - P_y = 1\), so

The area of a region can be found by walking around its boundary. For an ellipse \(x = a\cos t\), \(y = b\sin t\): \(\tfrac{1}{2}\int_0^{2\pi}(ab\cos^2 t + ab\sin^2 t)\,dt = \pi ab\). This is the principle of the planimeter, a mechanical instrument once used to measure areas on maps and indicator diagrams by tracing their outlines, and of the shoelace formula for polygons (MATH-125) used by surveying software today.

The divergence theorem

The divergence theorem (Gauss's theorem) is the 3D statement of conservation. For a solid region E with closed boundary surface S and outward normal \(\mathbf{n}\):

Here S is closed, so the surface integral covers the whole boundary. In words: the total flux out through the boundary equals the total of all the sources inside. Again, adjacent small boxes share faces where their fluxes cancel, leaving only the outer surface.

Gauss's law

For the electric field of a point charge, \(\mathbf{E} = \dfrac{q}{4\pi\varepsilon_0}\dfrac{\mathbf{r}}{|\mathbf{r}|^3}\), a direct calculation gives \(\nabla \cdot \mathbf{E} = 0\) everywhere except at the charge itself, and the flux through any sphere around the charge is \(q/\varepsilon_0\). Combining these with the divergence theorem shows that the flux through any closed surface that encloses the charge is \(q/\varepsilon_0\), whatever its shape: Gauss's law from PHYS-131. In differential form, \(\nabla \cdot \mathbf{E} = \rho/\varepsilon_0\).

Stokes' theorem

Stokes' theorem lifts Green's theorem off the plane. For an oriented surface S with boundary curve C, traversed so that the surface is on the left when you walk along C with your head in the direction of \(\mathbf{n}\) (the right-hand rule):

The circulation around the edge equals the total flux of curl through any surface spanning it. A remarkable consequence: every surface with the same boundary gives the same answer, so you can choose the simplest one.

Ampère's law

Stokes' theorem converts Ampère's law, \(\oint_C \mathbf{B} \cdot d\mathbf{r} = \mu_0 I_{enc}\), into its differential form, \(\nabla \times \mathbf{B} = \mu_0\mathbf{J}\) (for steady currents): circulation of the magnetic field around a loop equals \(\mu_0\) times the current passing through any surface bounded by the loop. Together with Gauss's law and Faraday's law, these are Maxwell's equations, the foundation of electrical engineering.

The big picture: conservation laws

Here is how the divergence theorem turns a physical balance into a differential equation, the template for many of the governing equations of engineering. Consider heat in a solid with density \(\rho\), specific heat \(c\), conductivity \(k\) and temperature \(T\). For any region E, the rate of increase of thermal energy inside equals the rate heat flows in through the boundary. Heat flux is \(\mathbf{q} = -k\nabla T\) (Fourier's law), so

Because this holds for every region E, the integrands must be equal:

This is the heat equation, solved in heat transfer courses and in every thermal simulation program. Replace heat by mass and you get the continuity equation; by momentum, the Navier-Stokes equations of fluid flow; by charge, Maxwell's equations. Multivariable calculus is the common language of all of them.