Foundations — Vorticity — ω = ∇ × v, circulation Γ

Before you can read the parent note Vorticity, you must own every symbol it fires at you. We build each one from nothing, in an order where every step only uses things already defined.

1. What is a "field"? (the picture underneath everything)

The bold letter $\mathbf{v}$ means a vector — a quantity with both a size and a direction, drawn as an arrow. A plain letter like $\Gamma$ (no bold) means a scalar — just one number, no direction.

Look at the figure: at each grid crossing sits a little arrow. Long arrows = fast flow, short = slow. This carpet of arrows is the velocity field. Everything in this topic is a question you ask about this carpet.

2. Components: cutting an arrow into $v_x$ and $v_y$

The little hats $\hat{x},\hat{y},\hat{z}$ are unit arrows — arrows of length exactly $1$ pointing along each axis. They are the "rulers" we measure direction against. Writing $\mathbf{v}=v_x\hat{x}+v_y\hat{y}+v_z\hat{z}$ just says: "take $v_x$ steps east, $v_y$ steps north, $v_z$ steps up." That sum lands you at the tip of the arrow.

3. The partial derivative $\frac{\partial }{\partial x}$ — "how fast does something change as I step sideways?"

This is the single most important tool the parent note assumes. We build it from zero.

Why this exact tool and not an ordinary derivative? An ordinary derivative $\frac{dv}{dx}$ works when $v$ depends on one variable. Our velocity depends on $x$, $y$, and $z$ all at once. The curly $\partial$ (say "partial") is the promise: "I am changing only $x$ and keeping the others locked." That is precisely what we need to detect spin — spin is about how the east velocity changes as you move north, versus how the north velocity changes as you move east.

In the figure: to get $\partial v_y/\partial x$ we compare the north-arrows at two points that differ only in their east position (red pair). To get $\partial v_x/\partial y$ we compare east-arrows at two points differing only in north position (amber pair). These two comparisons are the whole engine of vorticity.

4. Rotation rate $\Omega$ — the spin of a rigid thing

This links to Angular Velocity of Rigid Bodies: for a solid spinning disc every point shares the same $\Omega$. For a fluid, different regions can have different $\Omega$, so it becomes a field too. The parent note's headline result $\mathbf{\omega }=2\mathbf{\Omega }$ says the curl gives back twice this local turning rate — hold onto the factor of $2$.

5. The little bold omega $\mathbf{\omega }$ — vorticity as a vector

Why a vector and not a number? Because spin needs an axis — a paddle-wheel can be tilted to face any direction, and $\mathbf{\omega }$ records which way its pin points. In flat 2D flow the axis is always straight out of the page, so only the $z$-part ${\omega }_z$ survives, and we can treat it like a number.

6. The curl symbol $\nabla \times$ — the spin-detector

The $\times$ borrows the shape of the cross product — the operation that already means "give me a vector perpendicular to two others." Here it produces the spin-axis arrow. Full recipe (from the parent note), each bracket being one of our building-block slope differences: $\mathbf{\omega }=\nabla \times \mathbf{v}=\left(\frac{\partial v_z}{\partial y}-\frac{\partial v_y}{\partial z}\right)\hat{x}+\left(\frac{\partial v_x}{\partial z}-\frac{\partial v_z}{\partial x}\right)\hat{y}+\left(\frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y}\right)\hat{z}$

Why a difference of two slopes? Look back at figure s02. A pure spin makes the east-velocity climb as you go north and the north-velocity drop as you go east — the two slopes have opposite sign, so subtracting them adds their spins together. A pure stretch (no spin) makes them equal, so the subtraction cancels to zero. The minus sign is the filter that throws away non-spinning motion. Deepen this in Curl and Divergence (vector calculus).

The figure shows the three flavours of motion the curl must sort out: pure spin (both arms turn the same way → curl $\ne 0$), pure shear (one arm turns, curl still fires), and pure stretch (arms just lengthen, no turn → curl $=0$).

7. The loop integral ${\oint }_C\mathbf{v}\cdot d\mathbf{l}$ — circulation

Two new pieces of notation live here; we unpack both.

Put them together: $\Gamma ={\oint }_C\mathbf{v}\cdot d\mathbf{l}$ reads: "walk once around the loop $C$; at every tiny step, record how much the flow pushed you along your step; add it all up." That total is the circulation $\Gamma$ — units ${\text{m}}^2/\text{s}$ (velocity $\times$ length). It is the "total swirl felt around the fence."

In the figure, at each step the amber component is $\mathbf{v}\cdot d\mathbf{l}$: where the flow runs with your walk it adds a positive slice; against your walk, a negative slice. Sum every slice for $\Gamma$.

8. Area, flux, and the bridge $d\mathbf{A}$

This is the machinery of Stokes' Theorem, which welds sections 6 and 7 together: $\Gamma ={\oint }_C\mathbf{v}\cdot d\mathbf{l}={\iint }_S\mathbf{\omega }\cdot d\mathbf{A}.$ In words: the total push around the fence equals all the tiny spins added up inside it. Local spin (curl) and global swirl (circulation) turn out to be the same fact seen two ways.

Prerequisite map

Everything on the left is a foundation; the two arrows meeting at Stokes theorem are the payoff — see Irrotational Flow and Velocity Potential and Kelvin's Circulation Theorem for where it leads.

Equipment checklist