Multivariable calculus

Curl of the Vector Field (z, x, y)

The curl of the field with components z, x and y is the constant vector (1, 1, 1). In each difference one partial is 1 and the other is 0: the x-partial of the second component is 1 while the y-partial of the first is 0, and the same shape repeats twice. The field is also divergence free.

P(x,y,z)=zP(x,y,z) = z
Q(x,y,z)=xQ(x,y,z) = x
R(x,y,z)=yR(x,y,z) = y
∇×F=⟨1, 1, 1⟩\nabla \times \mathbf{F} = \langle 1,\ 1,\ 1 \rangle

Three differences, one live term each

Each component of this field is a single variable, and it is never its own. PP carries zz, QQ carries xx, RR carries yy. So in each curl slot exactly one of the two partials is 11 and the other is 00.

∂R∂y−∂Q∂z=1−0=1\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = 1 - 0 = 1
∂P∂z−∂R∂x=1−0=1\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} = 1 - 0 = 1
∂Q∂x−∂P∂y=1−0=1\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 1 - 0 = 1

Compare this with ⟨x2,y2,z2⟩\langle x^2, y^2, z^2 \rangle, where each component carries its own variable and the curl is zero. The difference between a large curl and no curl at all is only which variable sits in which slot.

A constant curl inside a field that is not constant

The field changes from point to point, but the spin does not. Every point in space carries the same curl vector, pointing along the diagonal direction ⟨1,1,1⟩\langle 1,1,1 \rangle with length 3\sqrt{3}.

∥∇×F∥=3,∇⋅F=0\lVert \nabla \times \mathbf{F} \rVert = \sqrt{3}, \qquad \nabla \cdot \mathbf{F} = 0

Physically: a paddle wheel whose axle points along 13⟨1,1,1⟩\frac{1}{\sqrt{3}}\langle 1,1,1 \rangle spins as fast as it can here, one with its axle perpendicular to that direction does not spin at all, and the answer is the same wherever you put it. The zero divergence says the flow neither compresses nor spreads.

The mistake: expecting the answer to contain a variable

Curl is built from first derivatives, so a field whose components are linear has a constant curl. Students who get ⟨1,1,1⟩\langle 1,1,1 \rangle often assume they lost a variable somewhere and go back through the algebra looking for it.

The related error is the reverse: seeing that no component contains its own variable and concluding that everything differentiates to zero. That confuses the curl with the divergence, which is genuinely zero here for exactly that reason.

Keep the two sorted by what they consume. Divergence eats the matched partials ∂P∂x\frac{\partial P}{\partial x}, ∂Q∂y\frac{\partial Q}{\partial y}, ∂R∂z\frac{\partial R}{\partial z}, which are all zero here. Curl eats the crossed ones, three of which equal 11.

Every component is checked numerically on each build, so the constant answer above is one the machine has reproduced from P, Q and R.

Frequently asked questions

Is this field conservative?

No. Its curl is ⟨1,1,1⟩\langle 1,1,1 \rangle, which is nonzero at every point, and a conservative field must have zero curl throughout. A search for a potential fails quickly: ∂f∂x=z\frac{\partial f}{\partial x} = z forces f=xz+g(y,z)f = xz + g(y,z), and then ∂f∂y=∂g∂y\frac{\partial f}{\partial y} = \frac{\partial g}{\partial y} cannot equal xx.

Which paddle wheel axis spins fastest?

The one pointing along the curl. For a unit vector n\mathbf{n}, the circulation per unit area about n\mathbf{n} is (∇×F)⋅n(\nabla \times \mathbf{F}) \cdot \mathbf{n}, which is largest when n\mathbf{n} is parallel to ⟨1,1,1⟩\langle 1,1,1 \rangle and zero when n\mathbf{n} is perpendicular to it, for instance n=12⟨1,−1,0⟩\mathbf{n} = \frac{1}{\sqrt{2}}\langle 1,-1,0 \rangle.