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The two fundamental theorems of three dimensions: circulation round a rim as the flux of the curl through any surface it bounds, and flux out of a closed surface as the divergence integrated over the solid inside.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to say which of the two theorems a given integral calls for, orient a boundary curve and its surface compatibly, compute a circulation as a flux of the curl and a flux as an integral of the divergence, swap one surface for another with the same rim, and say precisely which hypothesis fails when a field has a singularity inside.
Green's theorem, which traded a plane loop for the region inside it; surface integrals and flux; and the divergence and curl from the start of this unit, each introduced as a local measurement — outflow per unit volume, circulation per unit area. This lesson integrates those two sentences, and that is all either theorem is.
A surface is bounded by a curve when that curve is its edge; the two are compatibly oriented when the right hand's fingers curl along the curve and the thumb points along the normal. A surface is closed when it has no edge and encloses a solid — a sphere or the surface of a box — and its standard orientation is outward. A solid is simple when the divergence theorem's proof by slicing applies to it, which every region in this course is.
Stokes' theorem. Let $S$ be an oriented piecewise smooth surface bounded by a simple closed curve $C$, oriented compatibly, and let $\mathbf{F}$ have continuous partial derivatives on an open set containing $S$. Then
$$\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}.$$
The circulation round the edge equals the flux of the curl through the inside. With $S$ flat in the $xy$ plane this is Green's theorem, letter for letter.
The divergence theorem. Let $E$ be a solid bounded by a closed surface $S$ oriented outward, with $\mathbf{F}$ smooth on an open set containing $E$. Then
$$\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_E (\nabla \cdot \mathbf{F})\,dV.$$
The net escape through the boundary equals the total divergence inside.
They are one statement. So is Green's theorem, and so is the fundamental theorem of calculus: the integral of a derivative over a region equals an integral of the function over that region's boundary, one dimension down. $\int_a^b f' = f(b) - f(a)$ is the case where the region is an interval and its boundary is two points.
Two consequences worth carrying. Stokes says the flux of a curl through any two surfaces with the same boundary is the same, so an awkward surface may be swapped for a flat one. The divergence theorem says the flux of a curl out of any closed surface is zero, since a closed surface has no boundary curve — which is $\nabla \cdot (\nabla \times \mathbf{F}) = 0$ integrated.
Another way: picture
For Stokes, tile the surface with tiny loops. Neighbouring loops share an edge and walk it in opposite directions, so everything interior cancels and only the outer rim survives; each tiny loop's circulation is the curl times its area. For the divergence theorem, fill the solid with tiny boxes. Neighbouring boxes share a face and count its flux with opposite signs, so everything interior cancels and only the outer skin survives. The same cancellation twice, one dimension apart.
Another way: steps
To choose and use one of them:
Both theorems are equalities, so either side may be computed. The derivative side is usually easier, because a divergence or a curl is frequently constant and then the integral is a multiplication. The flux out of a box, computed directly, is six surface integrals; computed by the theorem it is often one product.
Stokes offers a second freedom that has no analogue elsewhere. Its right-hand side depends on $S$ only through its boundary, since the left-hand side mentions nothing else. So any surface with the same boundary curve gives the same answer: a hemisphere may be replaced by the flat disc that shares its rim, a dented drum head by a flat one. Choosing the flat replacement is often the entire trick in a problem that looks impossible.
The same reasoning run the other way gives the identity for free. A closed surface has no boundary curve at all, so the flux of any curl out of it is the circulation round an empty curve, namely zero — and by the divergence theorem that flux is the integral of $\nabla \cdot (\nabla \times \mathbf{F})$, which is therefore zero. An identity proved in the first lesson of this unit by cancelling second derivatives, proved again here by a picture.
Mismatched orientations. Stokes pairs a direction round the curve with a side of the surface, by the right-hand rule. Getting them the wrong way round negates the answer while leaving it the right size, which is the hardest kind of error to notice.
Using an inward normal on a closed surface. The divergence theorem is stated with the outward orientation; inward gives the negative.
Applying Stokes to a surface that is not bounded by the curve given. The curve must be the whole edge. A hemisphere with a hole punched in it has two boundary circles, and both appear.
Checking smoothness on the boundary only. This is the same failure Green's theorem has, one dimension up, and it is the one that produces confident wrong answers. Green, Stokes and the divergence theorem each hold on a region of a particular shape, with a field defined everywhere inside it and a boundary oriented in a particular way. A theorem quoted where one of those fails has not been applied; it has been guessed with, and the answer it gives can be wrong by exactly the amount the missing hypothesis was carrying.
This is the same mistake Green's theorem punishes, and it is worth meeting twice because in three dimensions it is the mistake that physics is built on noticing. The inverse-square field is perfectly well behaved on any sphere centred at the origin, and its divergence is zero at every single point where it exists. The divergence theorem still does not apply, because one point of the solid — the centre — is not a point of the field's domain.
The consequence is not an error to be avoided but Gauss's law: the flux out of any closed surface enclosing a charge is the same number, regardless of the surface's shape or size, and zero for any surface that excludes it. The whole apparatus of electrostatics, and the reason a conductor's interior field vanishes, is this one hypothesis failing in a controlled way.
So the habit to carry out of this course: before quoting either theorem, name where the field fails to exist, and ask whether any of those places is inside.
Find $\oint_C \mathbf{F} \cdot d\mathbf{r}$ for $\mathbf{F} = \langle -y, x, z^2 \rangle$ round the unit circle in the plane $z = 0$, counterclockwise seen from above.
A closed curve bounding a surface: Stokes.
The curl is $\langle 0, 0, 2 \rangle$, and the compatible orientation for a counterclockwise rim is the upward normal, so the integrand is $2$.
The right-hand rule fixes the normal.
Taking $S$ to be the flat disc, the flux is $2$ times its area $\pi$, so the circulation is $2\pi$. The $z^2$ component never appeared: it contributes nothing to the curl, and a field's irrelevant parts drop out of the derivative side automatically.
The easy surface with the same rim.
Find the outward flux of $\mathbf{F} = \langle x, y, z \rangle$ through the surface of the cube $0 \le x,y,z \le 2$.
A closed surface bounding a solid: the divergence theorem.
The divergence is $1 + 1 + 1 = 3$, constant, and the cube has volume $8$.
A constant divergence needs no limits.
So the flux is $24$. Directly this is six integrals; and the answer makes sense, since this field points straight out everywhere and grows with distance, so every face leaks. A theorem that also explains its answer is worth more than one that only produces it.
Three times the volume.
Let $\mathbf{G} = \nabla \times \mathbf{F}$ for some smooth $\mathbf{F}$, and let $S$ be any closed surface. Find the outward flux of $\mathbf{G}$ through $S$.
A closed surface suggests the divergence theorem.
By the divergence theorem the flux is $\iiint_E \nabla \cdot \mathbf{G} \,dV$, and $\nabla \cdot (\nabla \times \mathbf{F}) = 0$ at every point.
So the flux is $0$, whatever $\mathbf{F}$ and whatever $S$. Stokes says the same thing from the other side: a closed surface has no boundary curve, so the circulation it would equal is an integral round nothing. Two theorems, one answer, and a third proof of an identity first met by differentiating — which is about as much agreement as this subject ever offers.
Match each integral to the theorem that rewrites it. Take every surface here to have area $7$.
| Stokes: the flux of the curl through the surface | The divergence theorem: the divergence over the solid | Green: a double integral over the region enclosed | The fundamental theorem of calculus: a difference of two values | |
|---|---|---|---|---|
| $\oint_C \mathbf{F} \cdot d\mathbf{r}$ round a space curve bounding a surface | ||||
| $\iint_S \mathbf{F} \cdot d\mathbf{S}$ over a closed surface | ||||
| $\oint_C \mathbf{F} \cdot d\mathbf{r}$ round a curve in the plane | ||||
| $\int_a^b f'(x)\,dx$ |
For $\mathbf{F} = \langle x, 5y, 3z \rangle$ and the box $0 \le x \le 2$, $0 \le y \le 2$, $0 \le z \le 4$, fill in the table.
| Value | |
|---|---|
| The divergence of the field | |
| The volume of the box | |
| The outward flux through the whole surface |
Use Stokes' theorem to find the counterclockwise circulation of $\mathbf{F} = \langle -5y,\; 5x,\; 0 \rangle$ round the boundary of the square $0 \le x \le 4$, $0 \le y \le 4$ in the plane $z = 0$.
Answer:
Find the outward flux of $\mathbf{F} = \langle 3x, 0, 0 \rangle$ through the whole surface of the cube $0 \le x, y, z \le 4$.
Answer:
Match each integral to the theorem that rewrites it. Take every surface here to have area $3$.
| Stokes: the flux of the curl through the surface | The divergence theorem: the divergence over the solid | Green: a double integral over the region enclosed | The fundamental theorem of calculus: a difference of two values | |
|---|---|---|---|---|
| $\oint_C \mathbf{F} \cdot d\mathbf{r}$ round a space curve bounding a surface | ||||
| $\iint_S \mathbf{F} \cdot d\mathbf{S}$ over a closed surface | ||||
| $\oint_C \mathbf{F} \cdot d\mathbf{r}$ round a curve in the plane | ||||
| $\int_a^b f'(x)\,dx$ |
The inverse-square field has divergence zero at every point where it is defined, yet its outward flux through a sphere of radius $6$ centred at the origin is $4\pi$. What follows?
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
For $\mathbf{F} = \langle 3x, 5y, 5z \rangle$ and the box $0 \le x \le 2$, $0 \le y \le 2$, $0 \le z \le 2$, fill in the table.
| Value | |
|---|---|
| The divergence of the field | |
| The volume of the box | |
| The outward flux through the whole surface |
You can state both theorems, match each to the integral it rewrites, and name the hypothesis that fails when a field is undefined inside. Look back over the whole subject: vectors gave you the geometry of space, vector functions the motion along a curve, partial derivatives the gradient and its uses, multiple integrals the totals over regions — and these last two theorems tie all four together into one sentence about derivatives and boundaries.
10. Your turn: the flux of a curl out of a closed surface, step 3