Vector Calculus

1. Vector fields

Picture every point in space carrying a little arrow. Standing in a river, you feel the current — its direction and speed depend on where you are. A weather map's wind arrows are the same idea: at each location, an arrow. That assignment — point in, arrow out — is a vector field, and almost everything in vector calculus is a question about one: how much of it flows through a surface (flux), how much it swirls around a point (curl), how much it spreads out from a source (divergence). It's the language physicists reach for whenever they describe fluid flow, electromagnetism, or heat diffusion.

The mental picture matters. Wind across a map, water flowing in a stream, the electric field around a charge, the gravitational pull near a planet — each of these is a rule that says "at this point, here's the local direction and strength." A vector field is that rule made precise.

From an algebraic field you can extract two scalars and one vector by differentiating: how much the field is spreading out at a point (divergence), how much it is swirling at a point (curl), and — if the field happens to be the gradient of something — what the underlying scalar landscape is.

2. The three differential operators

One symbol — the nabla $\nabla = \langle \partial_x,\ \partial_y,\ \partial_z \rangle$ — does triple duty. The three operations differ only in what kind of object you feed it and which product you use.

Gradient — already met

For a scalar field $f$, the gradient turns it into a vector field:

It points in the direction of steepest increase of $f$, and its magnitude is the rate of that increase. We covered this in Partial derivatives; here it returns as the bridge between scalar landscapes and vector fields.

Divergence — a scalar

Imagine a tiny box centered at the point. Divergence measures the net outflow per unit volume in the limit as the box shrinks. A field with $\nabla \cdot \mathbf{F} > 0$ at $\mathbf{p}$ acts like there's a tiny faucet at $\mathbf{p}$; with $\nabla \cdot \mathbf{F} < 0$ it acts like a drain. Solenoidal fields ($\nabla \cdot \mathbf{F} = 0$ everywhere) are incompressible — what enters any region also leaves.

Example: $\mathbf{F}(x,y,z) = \langle x, y, z \rangle$ (the field that radiates straight out from the origin). Then $\nabla \cdot \mathbf{F} = 1 + 1 + 1 = 3$ — a uniform source everywhere.

Curl — a vector

The determinant mnemonic captures the same formula:

Example: $\mathbf{F}(x,y,z) = \langle -y, x, 0 \rangle$ (the field of a rigid rotation around the $z$-axis). Then $\nabla \times \mathbf{F} = \langle 0, 0, 2 \rangle$ — pointing up the rotation axis, with magnitude $2$ (twice the angular speed of $1$). Irrotational fields ($\nabla \times \mathbf{F} = \mathbf{0}$) have no local swirling; on a simply connected region they are exactly the gradient fields.

One picture, two readings

Divergence and curl are independent: a field can have any combination of the two. The gradient $\nabla f$ is the third leg — it's the operator that builds a vector field from a scalar. Two identities tie them together and you will eventually rely on them without thinking:

In words: the curl of a gradient is zero (gradient fields don't swirl), and the divergence of a curl is zero (swirl fields don't have sources). Both follow from equality of mixed partials.

3. Line integrals and work

Now we add up. The simplest thing you can add up along a curve is a vector field's component along the direction of travel — the work it does on a particle moving through it.

This is what physicists call work. If $\mathbf{F}$ is a force field and $\mathbf{r}(t)$ is the path of a particle, $\int_C \mathbf{F} \cdot d\mathbf{r}$ is the work done by the force over the path. Push a box across a frictionless floor in the direction the force is pushing: positive work. Drag it sideways to the push: zero work. Pull against the force: negative work.

The notation $d\mathbf{r}$ is shorthand for $\mathbf{r}'(t)\, dt$ — a tiny tangent displacement. Two consequences fall out:

• The integral is independent of the parametrisation as long as orientation is preserved.
• Reversing the direction of travel negates the integral: $\int_{-C} \mathbf{F} \cdot d\mathbf{r} = -\int_C \mathbf{F} \cdot d\mathbf{r}$.

There is also a scalar line integral $\int_C f\, ds$ — adding a scalar weighted by arc length, no dot product, no direction. It's used for masses of wires and average values along a curve, but the vector version is the one that drives the rest of the chapter.

4. Conservative fields and the FTC for line integrals

Sometimes a vector field is itself the gradient of some scalar: $\mathbf{F} = \nabla f$. Such a field is called conservative, and the scalar $f$ is its potential. Gravity and electrostatics both produce conservative fields — the potential energy you remember from physics is exactly $f$.

This is the same shape as the original fundamental theorem of calculus — $\int_a^b f'(x)\, dx = f(b) - f(a)$ — promoted one dimension up. The "derivative" is the gradient; the "boundary" of a curve is its two endpoints (with signs); the integral of the derivative over the curve equals the values of the function on the boundary.

How to tell whether a field is conservative

Compute the curl. If $\nabla \times \mathbf{F} = \mathbf{0}$ and the domain is simply connected (no holes you can't shrink away), then $\mathbf{F}$ is conservative. To actually find the potential $f$, integrate one component at a time, using the next component to pin down the integration constant.

Quick example: $\mathbf{F} = \langle 2x, 2y \rangle$. The curl (in 2D, the scalar $Q_x - P_y$) is $0 - 0 = 0$, so a potential exists; integrating $f_x = 2x$ gives $f = x^2 + g(y)$, and matching $f_y = 2y$ forces $g'(y) = 2y$, so $f = x^2 + y^2$ (up to a constant). The work from $(0,0)$ to $(1,1)$ along any path is $f(1,1) - f(0,0) = 2$.

5. Surface integrals and flux

For surfaces, the natural thing to add up is the part of a vector field that pokes through perpendicularly — the flux. Picture water flowing past a fishing net: flux measures how much water passes through per unit time.

For a closed surface — like a sphere — the conventional choice is the outward normal, and flux measures net outflow: how much more is leaving than entering. That orientation is what makes the divergence theorem read so cleanly.

As with line integrals, there's also a scalar surface integral $\iint_S f\, dS$, used for areas and centres of mass. Flux is the one that drives the theorems.

6. One big idea: boundary = bulk of derivative

Now the magic. Three different-looking theorems — Green's in the plane, Stokes' on surfaces, the divergence theorem in space — are all expressing the same sentence:

Integrating a field along the boundary of a region equals integrating its derivative throughout the region itself.

You've already seen this idea twice. The fundamental theorem of calculus says $\int_a^b f'(x)\, dx = f(b) - f(a)$: bulk integral of the derivative on the left, "boundary values" on the right. The FTC for line integrals does the same for gradients along a path. The next three theorems do it for the other two operators — curl and divergence — in higher dimensions.

In differential-forms language all three (and the FTC) collapse into one statement, the generalised Stokes' theorem: $\int_{\partial \Omega} \omega = \int_\Omega d\omega$. Keep that as a slogan — but the three concrete cases below are what you'll actually use.

The proofs all follow the same physical intuition: cut the region into infinitesimal pieces, observe that adjacent pieces share boundaries with opposite orientations so internal contributions cancel, and only the outer boundary survives. We'll see the picture for Green's; the others are the same idea in higher dimensions.

7. Green's theorem

Let $D$ be a region in the plane whose boundary $C = \partial D$ is a piecewise-smooth simple closed curve, oriented positively (counterclockwise, so the region stays on your left). For continuously differentiable $P(x,y)$ and $Q(x,y)$:

The integrand on the right is the 2D version of curl: it measures how much the field $\langle P, Q \rangle$ would spin a paddlewheel placed at that point.

The proof picture

Chop $D$ into a fine grid of tiny rectangles. For each rectangle, the boundary integral $\oint P\, dx + Q\, dy$ around it is approximately $(Q_x - P_y)\, dA$ — that's literally the definition of those partials. Now sum over rectangles: every internal edge appears in two adjacent rectangles with opposite orientations, so the contributions cancel exactly. Only the outer boundary $C$ survives on the left, and the sum of $(Q_x - P_y)\, dA$ becomes the double integral on the right.

Canonical example: area as a boundary integral

Choose $P = -y/2$ and $Q = x/2$. Then $Q_x - P_y = 1/2 + 1/2 = 1$, and Green's theorem gives

The area of a region is encoded entirely in a single integral around its boundary. For a circle of radius $R$ parametrised as $\mathbf{r}(t) = (R\cos t, R\sin t)$:

This is how planimeters — the mechanical devices surveyors used to trace map outlines and read off the enclosed area — work physically.

8. Stokes' theorem

Green's theorem promoted to three dimensions. Let $S$ be a piecewise-smooth oriented surface in $\mathbb{R}^3$ whose boundary $\partial S$ is a simple closed curve $C$, oriented compatibly with $S$ (right-hand rule: curl your fingers along $C$, thumb points along the chosen normal of $S$).

Circulation around the boundary equals total flux of curl through the surface. The left side is what a stick floating on a swirling fluid would feel along the rim; the right is the sum of all the tiny swirls inside the surface. Once again, adjacent infinitesimal circulations cancel, leaving only the boundary.

A surprising corollary: the flux of curl depends only on the boundary, not on which surface you stretch across that boundary. You can integrate over a hemisphere, a flat disk, or a stretched-out balloon — same answer, as long as the rim is the same.

Canonical example: curl through the unit disk

Take $\mathbf{F} = \langle -y, x, 0 \rangle$ and let $S$ be the unit disk in the $xy$-plane, oriented with $\hat{\mathbf{n}} = \hat{\mathbf{k}}$. Its boundary $C$ is the unit circle, parametrised counterclockwise as $\mathbf{r}(t) = (\cos t, \sin t, 0)$ for $t \in [0, 2\pi]$.

Boundary side:

Surface side: $\nabla \times \mathbf{F} = \langle 0, 0, 2 \rangle$, and $(\nabla \times \mathbf{F}) \cdot \hat{\mathbf{k}} = 2$, so

The two sides match, as the theorem promises.

9. The divergence theorem

The same idea, one dimension up again. Let $E$ be a solid region in $\mathbb{R}^3$ with piecewise-smooth boundary surface $\partial E = S$, oriented with the outward normal. For continuously differentiable $\mathbf{F}$:

Net flux out of a closed surface equals the total source strength inside. If the field is solenoidal ($\nabla \cdot \mathbf{F} = 0$), no net flow crosses any closed surface — what comes in goes out.

The proof idea mirrors Green's: chop the solid into infinitesimal boxes; each box's flux is approximately $(\nabla \cdot \mathbf{F})\, dV$; faces shared between adjacent boxes have opposite outward normals and cancel; only the true outer surface survives.

Canonical example: outward flow through the unit ball

Take $\mathbf{F} = \langle x, y, z \rangle$ and let $E$ be the closed unit ball, $S$ the unit sphere with outward normal $\hat{\mathbf{n}} = \mathbf{F}/|\mathbf{F}| = \mathbf{F}$ on the surface where $|\mathbf{F}| = 1$.

Volume side: $\nabla \cdot \mathbf{F} = 3$ everywhere, so

Surface side: $\mathbf{F} \cdot \hat{\mathbf{n}} = |\mathbf{F}| = 1$ on the unit sphere, so

Both sides give $4\pi$. The physical reading: the radial field $\langle x, y, z \rangle$ has a uniform source density of $3$ throughout the ball; that source density piled up inside the ball exactly matches what's pouring out through its surface.

10. Common pitfalls