Divergence & Curl: Conservative, Solenoidal & Irrotational Vector Fields

The divergence of a vector field F = <P, Q, R> is a scalar quantity obtained by taking the sum of the partial derivatives of each component with respect to its own variable. Specifically, we compute the partial derivative of P with respect to x, add the partial derivative of Q with respect to y, and finally add the partial derivative of R with respect to z. This gives the standard formula div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z.

The curl is defined as the vector cross product of the del operator ∇ and the vector field F. The notation ∇ · F is for divergence, and ∇² is the Laplacian operator.

Divergence is a scalar that tells how much the vector field is expanding or compressing at a point. Positive divergence means the field is spreading out (acting like a source), negative divergence means it is converging (acting like a sink), and zero divergence means the field is incompressible at that point. This is directly related to the net amount of "flow" leaving a tiny closed surface around the point.

Curl is a vector that points in the direction of the axis of rotation and has magnitude equal to twice the angular velocity of the local rotation of the field. If you place a tiny paddle wheel at the point, the curl tells you in which direction and how fast it would spin due to the field.

We take the partial derivative of each component. ∂/∂x(cos x) = -sin x. ∂/∂y(sin y) = cos y. ∂/∂z(z²) = 2z. Summing these gives the divergence: -sin(x) + cos(y) + 2z.

Compute each component: i (∂R/∂y − ∂Q/∂z) = i (0 − 0) = 0; j component = − (∂R/∂x − ∂P/∂z) = − (0 − 0) = 0; k component = ∂Q/∂x − ∂P/∂y = ∂/∂x (x) − ∂/∂y (-y) = 1 − (−1) = 1 + 1 = 2. Thus curl F = <0, 0, 2>, a constant vector in the z-direction.

One of the standard vector calculus identities is that the divergence of the curl of any vector field is always zero. This is true because when you expand div(curl F) using the component formulas, all terms cancel out pairwise due to equality of mixed partial derivatives (Clairaut's theorem).

First, find the general formula for the divergence: ∂(x²)/∂x + ∂(xy)/∂y + ∂(z)/∂z = 2x + x + 1 = 3x + 1. Now, evaluate this expression at x = 1. Div F = 3(1) + 1 = 4. Note that the values of y and z do not affect the result in this specific case.

Away from the z-axis, this is the standard 2D rotation field extended constantly in z. Computing component-wise off the axis gives curl F = <0, 0, 0>. On the z-axis the field is undefined, but in the domain where it is defined, the curl is zero (though the circulation around a loop enclosing the origin is nonzero, showing it is not conservative globally).

div V = ∂/∂x (x) + ∂/∂y (y) + ∂/∂z (-2z) = 1 + 1 + (-2) = 0 everywhere.

We compute the partial derivatives: ∂/∂x (e^y) = 0 (since y is constant w.r.t x). ∂/∂y (e^z) = 0 (since z is constant w.r.t y). ∂/∂z (e^x) = 0 (since x is constant w.r.t z). The divergence is 0 + 0 + 0 = 0.

The velocity field for rigid rotation is v = ω × r = <-ω y, ω x, 0>. Now compute curl v: i (∂/∂y (0) − ∂/∂z (ω x)) = 0; j − (∂/∂x (0) − ∂/∂z (−ω y)) = 0; k (∂/∂x (ω x) − ∂/∂y (−ω y)) = ω − (−ω) = 2ω. So curl v = <0, 0, 2ω>, twice the angular velocity vector.

For a fluid in solid-body rotation, every particle moves in circles with the same angular velocity ω. The velocity field is v = <-ω y, ω x, 0>. As shown in the previous calculation, curl v = <0, 0, 2ω>, which is constant throughout the fluid and directed along the axis of rotation with magnitude twice the angular velocity.

P = yz, Q = xz, R = xy. i component: ∂R/∂y − ∂Q/∂z = ∂/∂y (xy) − ∂/∂z (xz) = x − x = 0. j component: − (∂R/∂x − ∂P/∂z) = − (y − y) = 0. k component: ∂Q/∂x − ∂P/∂y = z − z = 0. All components are zero, so curl F = 0.

Away from the origin, direct computation gives curl F = <0, 0, 0>. However, when you integrate curl F over any surface enclosing the origin (or use Stokes' theorem on a loop around the origin), the circulation is 2π, implying that curl F contains a Dirac delta function in the z-direction at the origin. This is the vector calculus description of an idealized point vortex.