Vector Calculus
Maxwell's four equations are written in one symbol — the inverted triangle ∇. This chapter unpacks it into gradient, divergence and curl, and meets the two great theorems that swap a field's behaviour at every point for its behaviour on a boundary.
- The three field integrals — line (circulation/work), surface (flux), and volume.
- The del operator \(\nabla\) and its three uses: gradient, divergence, curl.
- Gradient — the direction and rate of steepest increase of a scalar field.
- Divergence — the net outflow per unit volume — and the divergence theorem.
- Curl — the circulation per unit area — and Stokes' theorem.
- The Laplacian, and how to tell a solenoidal field from an irrotational one.
Integrals of Fields
Three integrals recur throughout electromagnetics. The line integral adds up the component of a field along a path — it gives work done by a force, or voltage along a route. The surface integral adds up the field passing through a surface — this is flux. The volume integral sums a quantity throughout a region, such as total charge from a charge density.
A line integral around a closed loop, written \(\oint_L \vec{A}\cdot d\vec{l}\), is the field's circulation. A surface integral over a closed surface is the net flux out of the volume it encloses. These two closed integrals are exactly what the divergence and Stokes theorems will connect to the del operator.
The Del Operator
The del (or nabla) operator \(\nabla\) is a vector whose components are partial-derivative instructions. In Cartesian coordinates:
By itself \(\nabla\) does nothing; it waits for something to act on. Applied three different ways it produces the three operators of this chapter: on a scalar it gives the gradient (\(\nabla V\)); dotted into a vector it gives the divergence (\(\nabla\cdot\vec{A}\)); crossed into a vector it gives the curl (\(\nabla\times\vec{A}\)).
Gradient of a Scalar
The gradient of a scalar field \(V\) is a vector that points in the direction of steepest increase of \(V\), with magnitude equal to that maximum rate of change. Think of \(V\) as elevation on a hill: \(\nabla V\) points straight uphill, and is longest where the slope is steepest.
The change in \(V\) along any step \(d\vec{l}\) is the gradient dotted with that step. Along a surface of constant \(V\) the change is zero, so \(\nabla V\) must be perpendicular to it. This is exactly why \(\vec{E} = -\nabla V\) is perpendicular to equipotentials in Chapter 6.
Divergence & Its Theorem
The divergence of a vector field measures the net outflow per unit volume at a point — whether that point acts as a source (positive divergence), a sink (negative), or neither. It is a scalar:
The divergence theorem (Gauss's theorem) is one of the two pillars of the course. It says the total outflow through a closed surface equals the sum of all the little outflows inside — the volume integral of the divergence:
This is the bridge between the integral and differential forms of Gauss's law in Chapter 5. It lets you trade a surface flux you can picture for a volume integral you can compute.
Curl & Stokes' Theorem
The curl of a vector field measures its circulation per unit area — how much the field swirls around a point. It is itself a vector, pointing along the axis of rotation by the right-hand rule. In Cartesian form it is a determinant:
Stokes' theorem is the second pillar. It says the circulation of a field around a closed loop equals the flux of its curl through any surface bounded by that loop:
This converts Ampère's and Faraday's laws between their integral and differential forms in Part 4. Where the divergence theorem trades a closed surface for a volume, Stokes' theorem trades a closed loop for the surface it rims.
The Laplacian
Applying divergence to a gradient gives the Laplacian of a scalar — the divergence of the steepest-ascent field. It measures how a point's value compares with the average of its neighbours, and it governs static potentials:
When \(\nabla^2 V = 0\) the field is harmonic — this is Laplace's equation, the heart of Chapters 9–10. Two null identities are worth memorising now, because they shape Maxwell's equations: the curl of a gradient is always zero, and the divergence of a curl is always zero.
Classifying Fields
The two null identities let us sort every vector field by what \(\nabla\) does to it. A field with zero divergence has no sources or sinks — its flux lines close on themselves; a field with zero curl does not swirl — it can be written as the gradient of a potential.
| Field type | Condition | Meaning | EM example |
|---|---|---|---|
| Solenoidal | \(\nabla\cdot\vec{A}=0\) | No source/sink; flux lines close | Magnetic field \(\vec{B}\) |
| Irrotational | \(\nabla\times\vec{A}=0\) | No swirl; has a scalar potential | Static electric field \(\vec{E}\) |
| Harmonic | \(\nabla^2 V=0\) | Value equals neighbour average | Potential in charge-free space |
Maxwell's equations answer exactly these for \(\vec{E}\) and \(\vec{B}\): their divergences give the source laws (Gauss), their curls give the circulation laws (Faraday, Ampère). The operators you learn here are the language those four equations are spoken in.
Worked Examples
Problem. Find \(\nabla V\) for \(V = x^2 y + yz\).
Solution. Differentiate with respect to each coordinate:
Problem. Find \(\nabla\cdot\vec{A}\) for \(\vec{A} = x^2\,\hat{a}_x + yz\,\hat{a}_y + 2z\,\hat{a}_z\) at \((1,1,1)\).
Solution. Add the three partial derivatives:
Problem. Find \(\nabla\times\vec{A}\) for \(\vec{A} = y\,\hat{a}_x - x\,\hat{a}_y\).
Solution. Only the \(z\)-component of the determinant survives:
Problem. For \(\vec{A} = x\,\hat{a}_x\) over the unit cube \(0\le x,y,z\le 1\), check \(\oint_S \vec{A}\cdot d\vec{S} = \int_v \nabla\cdot\vec{A}\,dv\).
Solution. The flux leaves only through the face at \(x=1\); the divergence is constant:
Problem. Show \(\vec{A} = y\,\hat{a}_x + x\,\hat{a}_y\) is irrotational and find its potential.
Solution. The curl vanishes, so a scalar \(V\) exists with \(\vec{A}=\nabla V\):
Problem. Verify that \(V = x^2 - y^2\) satisfies Laplace's equation.
Solution. Sum the second derivatives:
Chapter Summary
Line = circulation/work, surface = flux, volume = total quantity. Closed versions feed the big theorems.
\(\nabla\) acts three ways: gradient on a scalar, divergence (dot) and curl (cross) on a vector.
\(\nabla V\) points uphill, perpendicular to equipotentials; \(dV=\nabla V\cdot d\vec{l}\).
Net outflow per volume; \(\oint_S\vec{A}\cdot d\vec{S}=\int_v(\nabla\cdot\vec{A})\,dv\).
Circulation per area; \(\oint_L\vec{A}\cdot d\vec{l}=\int_S(\nabla\times\vec{A})\cdot d\vec{S}\).
Solenoidal (\(\nabla\cdot\vec{A}=0\)), irrotational (\(\nabla\times\vec{A}=0\)), harmonic (\(\nabla^2V=0\)).
Problems
For each item, identify whether it calls for gradient, divergence, curl, or a theorem, then apply it. Difficulty rises down the list.
- Find \(\nabla V\) for \(V = e^{-x}\sin y + z^2\).
- Find \(\nabla\cdot\vec{A}\) for \(\vec{A} = xy\,\hat{a}_x + y^2\,\hat{a}_y + xz\,\hat{a}_z\).
- Find \(\nabla\times\vec{A}\) for \(\vec{A} = z\,\hat{a}_x + x\,\hat{a}_y + y\,\hat{a}_z\).
- Compute \(\nabla^2 V\) for \(V = x^2 + y^2 + z^2\) and state whether it is harmonic.
- Evaluate the line integral of \(\vec{A} = x\,\hat{a}_x + y\,\hat{a}_y\) from \((0,0)\) to \((1,1)\) along the straight path.
- Verify Stokes' theorem for \(\vec{A} = y\,\hat{a}_x\) over the unit square in the \(xy\)-plane.
- Verify the divergence theorem for \(\vec{A} = r\,\hat{a}_r\) over a sphere of radius \(a\) (spherical divergence \(\nabla\cdot\vec{A} = \tfrac{1}{r^2}\tfrac{\partial}{\partial r}(r^2 A_r)\)).
- Show that any field of the form \(\vec{A} = \nabla V\) has zero curl, for arbitrary smooth \(V\).
- Determine whether \(\vec{A} = \rho\,\hat{a}_\phi\) is solenoidal (use the cylindrical divergence formula).
- Find the directional derivative of \(V = x^2 y\) at \((1,2,0)\) in the direction \(\hat{a}_x + \hat{a}_y\).
- Prove the identity \(\nabla\cdot(\nabla\times\vec{A}) = 0\) by direct expansion in Cartesian coordinates.
- A field is both solenoidal and irrotational in a region. What partial-differential equation must its potential satisfy, and why?