Foundations — Kelvin's circulation theorem

This page assumes nothing. Every arrow, symbol, and squiggle in the parent note Kelvin's circulation theorem is unpacked here, in an order where each idea leans only on the ones before it.

1. A fluid, and the velocity field $\mathbf{u}$

The picture: imagine a river photographed from above, with a tiny arrow glued to every point of the water surface. Long arrows = fast flow; the arrow's direction = where that bit of water is heading.

• Bold letters like $\mathbf{u}$ mean a vector — a thing with both size and direction (an arrow).
• Plain letters like $t$ or $p$ mean a scalar — just a number, no direction.

Why the topic needs it: Kelvin's theorem is a statement about how arrows arranged around a loop behave over time. Without the field of arrows, there is nothing to integrate.

2. A closed curve $C$ and the line element $d\mathbf{\ell }$

The picture: a rubber band lying on the water. Zoom into any point and the band looks like a straight tiny arrow — that arrow is $d\mathbf{\ell }$.

• The symbol ${\oint }_C$ means "add up something all the way around the closed loop $C$." The little circle on the integral sign is a reminder the path is closed.

Why the topic needs it: we will walk around a loop and, at each tiny step, ask "is the fluid flowing with me or against me?" To ask that, we need the step arrows $d\mathbf{\ell }$.

3. The dot product $\mathbf{u}\cdot d\mathbf{\ell }$

We need a tool that answers a specific question: "How much of the fluid's velocity points along my walking step?" The dot product is exactly that tool — that is why it appears and not, say, multiplication of lengths.

Reading the $\cos \theta$:

• If the arrows point the same way ($\theta =0$, $\cos 0=1$): the product is biggest and positive.
• If they are perpendicular ($\theta =90^{\circ }$, $\cos 90^{\circ }=0$): the product is zero — flow across your step counts for nothing.
• If they point opposite ($\theta =180^{\circ }$, $\cos 180^{\circ }=-1$): the product is negative — you are walking against the current.

Why the topic needs it: $\mathbf{u}\cdot d\mathbf{\ell }$ is large and positive exactly where the fluid runs along the loop the way we traverse it. That is the raw ingredient of "swirl."

4. Circulation $\Gamma$ — adding the swirl all around

The picture: if the fluid circles the loop like water spiralling down a drain, every step contributes a positive bit and $\Gamma$ is big. If the fluid just slides straight past the loop, the "with me" bits on one side cancel the "against me" bits on the other, and $\Gamma =0$.

Why the topic needs it: $\Gamma$ is the conserved quantity. Everything else is machinery to prove $\Gamma$ doesn't change.

5. Vorticity $\mathbf{\omega }$ and the curl $\nabla \times \mathbf{u}$

Circulation is about a whole loop. Vorticity is the same idea shrunk to a single point: "how fast is the fluid spinning right here?"

The picture: drop a tiny paddle-wheel into the flow. If it spins, $\mathbf{\omega }$ is nonzero there; the faster it spins, the longer the arrow; the axle direction is $\mathbf{\omega }$'s direction.

• $\nabla$ ("nabla" or "del") is a bundle of derivatives — a "how fast is this changing in space?" operator. Here it appears in the cross-product form $\nabla \times$; later we meet it as $\nabla p$ ("gradient of pressure").
• The cross product $\times$ of two arrows gives a third arrow perpendicular to both; the curl reuses that geometry to pull out a rotation axis.

Why the topic needs it: Kelvin's theorem is usually summed up as "vorticity is conserved along material loops." Vorticity is the local view; circulation is the loop view. The next section links them.

Deeper details live in Vorticity and the vorticity equation.

6. Stokes' theorem — the bridge

Reading the new symbols:

• $S$ is any surface whose rim is the loop $C$ — think of a soap film spanning the rubber band.
• ${\iint }_S$ means "add up over the whole surface" (a double sum, since a surface is two-dimensional).
• $d\mathbf{A}$ is a tiny patch of that surface, written as an arrow pointing straight out of the patch, with length equal to the patch's area.
• $\mathbf{\omega }\cdot d\mathbf{A}$ therefore counts only the spin whose axis threads through the patch — the dot product again picking out the aligned part.

Why the topic needs it: it lets us swap between "circulation of the loop" and "vorticity flux through the loop" freely — and the worked example $\Gamma =\omega A$ is nothing but Stokes' theorem for a small flat patch. Full statement: Stokes' theorem.

7. The material derivative $\frac{D}{Dt}$

There are two ways to ask "how fast is something changing?":

• $\frac{\partial }{\partial t}$: sit at a fixed spot and watch the value change (Eulerian view).
• $\frac{D}{Dt}$: ride along with a fluid particle and watch the value change as you move (Lagrangian view).

The picture: a thermometer floating down a river. Even if the river's temperature map is frozen in time ($\partial /\partial t=0$), the thermometer's reading still changes as it drifts from a warm patch to a cold one — that is the $\mathbf{u}\cdot \nabla$ piece.

Why the topic needs it: the entire theorem is the single statement $\frac{D\Gamma }{Dt}=0$. The loop is material — made of the same particles — so we must differentiate while moving with them.

8. Pressure $p$, density $\rho$, and the gradient $\nabla p$

• Pressure $p$ (scalar): how hard the fluid pushes outward per unit area at a point.
• Density $\rho$ ("rho", scalar): mass packed into each unit of volume.
• Gradient $\nabla p$ (vector): an arrow pointing in the direction pressure increases fastest, with length equal to how steeply it rises. Fluid gets pushed down the pressure gradient, from high $p$ to low $p$ — hence the minus sign in $-\frac{1}{\rho }\nabla p$.

The picture: a hill whose height is pressure; $\nabla p$ is the steepest-uphill arrow; a ball (fluid parcel) rolls the opposite way, downhill.

Why the topic needs it: the Euler equation says a fluid parcel accelerates because of pressure gradients and body forces. Two of Kelvin's proof steps are entirely about making the $\frac{1}{\rho }\nabla p$ term vanish.

9. The Euler equation

Plain words: (acceleration of a parcel) = (push from pressure differences) + (push from body forces like gravity). Here $\Phi$ ("Phi", scalar) is the potential of a conservative body force $\mathbf{f}=-\nabla \Phi$, e.g. gravity gives $\Phi =gz$.

• "Inviscid" = frictionless; there is no viscosity term. Real, sticky fluids add one, and that is exactly what can break Kelvin's theorem.

Why the topic needs it: the proof replaces $D\mathbf{u}/Dt$ by this right-hand side, then kills each surviving piece with a perfect-differential argument. Full note: Euler equation for ideal fluids.

10. Perfect differentials and the closed-loop trick

This single idea does the heavy lifting three separate times in the proof.

The picture: hike a mountain trail that loops back to the car park. You climbed and descended, but your net altitude change is exactly zero. Altitude is $f$; each little climb/descent is $df$.

Why the topic needs it: each of these is a perfect differential, hence each loop integral is zero:

• $\oint \mathbf{u}\cdot d\mathbf{u}=\oint d(\frac{1}{2}|\mathbf{u}{|}^2)=0$ (the stretching term),
• $\oint \nabla \Phi \cdot d\mathbf{\ell }=\oint d\Phi =0$ (the body-force term),
• $\oint \frac{1}{\rho }\nabla p\cdot d\mathbf{\ell }=\oint dP=0$ (the pressure term, only if barotropic).

11. Barotropic vs baroclinic

Only a pure gradient can be a perfect differential $dP$; that is the whole reason step 5 of the proof works.

Why the topic needs it: it is assumption (3), and its failure is the most physically important loophole (weather fronts, sea breezes).

Prerequisite map

Equipment checklist

Use these to test if you are ready for the parent theorem — cover the right side and answer.