Divergence & Curl: Vector Identities, Product Rules & Physical Applications

Using the curl formula: i (∂R/∂y − ∂Q/∂z) − j (∂R/∂x − ∂P/∂z) + k (∂Q/∂x − ∂P/∂y). Here R = 5z (no y), so ∂R/∂y = 0; Q = 4y (no z), so ∂Q/∂z = 0 → i component = 0 − 0 = 0. Similarly, ∂R/∂x = 0, ∂P/∂z = 0 → j component = −(0 − 0) = 0. ∂Q/∂x = 0, ∂P/∂y = 0 → k component = 0 − 0 = 0. So curl F = <0, 0, 0>.

The divergence is the sum of partial derivatives.

d/dx(5x) = 5.

d/dy(0) = 0.

d/dz(0) = 0.

Total divergence = 5 + 0 + 0 = 5.

Using the right-hand rule, if you curl the fingers of your right hand in the direction of the rotation (counter-clockwise), your thumb points up, which corresponds to the positive z-direction (the vector k).

We compute the curl using the determinant method or the component formula. P = y, Q = 0, R = 0. The k-component is ∂Q/∂x - ∂P/∂y. Here, Q = 0 and P = y. So, ∂Q/∂x = ∂(0)/∂x = 0 and ∂P/∂y = ∂(y)/∂y = 1. Thus, the k-component is 0 - 1 = -1. The i and j components are zero. The result is < 0, 0, -1 >. This illustrates that a fluid moving in straight lines can still have a non-zero curl if the velocity varies perpendicular to the flow (shear), causing a paddle wheel to rotate clockwise.

For rigid-body rotation with angular velocity vector ω = <0, 0, ω>, the velocity field is v = ω × r = <−ω y, ω x, 0>. The curl of this velocity field is always <0, 0, 2ω>, regardless of position. Here ω = 5, so |curl v| = |2ω| = 10 (in units of s⁻¹). The factor of 2 comes from the standard result for rigid rotation: curl v = 2ω.

The derivative acts on both f and F, giving div(fF) = grad(f)·F + f div(F).

The correct formula for the k-component (or scalar value) of the curl for a 2D field is (partial of Q with respect to x) minus (partial of P with respect to y). The statement incorrectly uses a plus sign instead of a minus sign. This is derived from the determinant of the matrix used to calculate the cross product.

Operations must match input/output types. The divergence operator takes a vector and returns a scalar. The gradient operator takes a scalar and returns a vector. Therefore, grad(div F) takes Vector -> Scalar -> Vector. Conversely, div(grad f) takes Scalar -> Vector -> Scalar (this is the Laplacian).

A necessary condition for a vector field to be conservative (i.e., to have a potential function f such that F = grad f) is that its curl must be zero everywhere. If the curl is non-zero, the field possesses "circulation" and cannot be generated purely by a scalar potential.

Divergence measures the rate of outgoing flux per unit volume. For a gas flow V, a positive divergence implies expansion (density decrease), and negative divergence implies compression (density increase). The divergence of V = < u(x), 0, 0 > is simply du/dx. Therefore, for compression (negative divergence), du/dx must be negative. This physically means the gas in front is moving slower than the gas behind it, causing it to "pile up."

We calculate the curl of V = < P, Q, R > = < y², 0, 0 >. The k-component of the curl is dQ/dx - dP/dy. Here, dQ/dx = 0 and dP/dy = 2y. So the curl is < 0, 0, -2y >. For the curl to be the zero vector, each component must be zero. This requires -2y = 0, which implies y = 0. Thus, the curl is zero only along the x-axis.

First, find the divergence: d/dx(x³/3) + d/dy(y³/3) + d/dz(z³/3) = x² + y² + z². Now, take the gradient of this scalar: < d/dx(x²+...), d/dy(...+y²+...), d/dz(...) > = < 2x, 2y, 2z >.

For a field to be conservative, its curl must be the zero vector. Let's calculate the k-component of the curl for F = < y, 0, 0 >: dQ/dx - dP/dy. Here P=y and Q=0.

dQ/dx = 0.

dP/dy = 1.

k-component = 0 - 1 = -1.

Since the curl is < 0, 0, -1 > (not zero), the field is not conservative.