Vector Calculus

Vector calculus describes quantities that vary in space: temperature (a scalar field), fluid velocity and electric fields (vector fields). It appears in the Engineering Mathematics section of the CE, ME, EE and EC papers.

Gradient and directional derivative

For a scalar field φ(x, y, z), the gradient is ∇φ = (∂φ/∂x, ∂φ/∂y, ∂φ/∂z).

• ∇φ points in the direction of the fastest increase of φ, and its magnitude is that rate.
• ∇φ is normal to the level surface φ = constant; the unit normal is ∇φ/|∇φ|.
• The directional derivative in the direction of a unit vector â is D_aφ = ∇φ · â. Its maximum value is |∇φ|.

Example. φ = x²y + yz³ at (1, 2, 1): ∇φ = (2xy, x² + z³, 3yz²) = (4, 2, 6). In the direction of (1, 2, 2), the unit vector is (1, 2, 2)/3, so D = (4 + 4 + 12)/3 = 20/3.

Divergence and curl

For F = (F₁, F₂, F₃):

• Divergence ∇·F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z, a scalar: the net outflow per unit volume. F is solenoidal if ∇·F = 0.
• Curl ∇ × F = (∂F₃/∂y − ∂F₂/∂z, ∂F₁/∂z − ∂F₃/∂x, ∂F₂/∂x − ∂F₁/∂y), a vector: the rotation. F is irrotational (conservative) if ∇ × F = 0, and then F = ∇φ for a scalar potential φ.

Identities (always true for smooth fields):

• ∇ × (∇φ) = 0 (the curl of a gradient is zero)
• ∇·(∇ × F) = 0 (the divergence of a curl is zero)
• ∇·(∇φ) = ∇²φ (the Laplacian)
• ∇ × (∇ × F) = ∇(∇·F) − ∇²F
• For r = (x, y, z): ∇·r = 3, ∇ × r = 0, ∇r = r/r.

Line, surface and volume integrals

• Line integral ∫_C F·dr: the work done by F along C. For a conservative field it depends only on the end points: ∫ ∇φ·dr = φ(B) − φ(A), and it is zero around any closed curve.
• Surface integral ∬_S F·n̂ dS: the flux through S.
• Volume integral ∭_V f dV.

The three theorems

A handy special case of Green's theorem: the area of a region is ½∮(x dy − y dx).

Worked example

Check the directional derivative above, the divergence and curl of F = (x²y, yz, xz²) at (1, 1, 1), and verify Gauss's theorem for F = (x, y, z) over the unit cube (flux = ∭ 3 dV = 3).

def grad(f, p, h=1e-6):
    return [(f(*[p[j] + (h if j == i else 0) for j in range(3)]) - f(*[p[j] - (h if j == i else 0) for j in range(3)])) / (2 * h) for i in range(3)]

phi = lambda x, y, z: x * x * y + y * z ** 3
g = grad(phi, [1, 2, 1])
print("grad phi at (1, 2, 1):", [round(v, 6) for v in g])
a = [1 / 3, 2 / 3, 2 / 3]
print(f"directional derivative = {sum(gi * ai for gi, ai in zip(g, a)):.6f} (20/3 = {20 / 3:.6f})")

# F = (x^2 y, y z, x z^2): div = 2xy + z + 2xz; curl = (0 - y, 0 - z^2, 0 - x^2)
x, y, z = 1, 1, 1
print("div F at (1,1,1) =", 2 * x * y + z + 2 * x * z, " curl F =", (-y, -z * z, -x * x))

# Gauss: outward flux of F = (x, y, z) through the six faces of the unit cube, by a midpoint sum
F = lambda p: p
n = 50
flux = 0.0
for axis in range(3):
    for side, normal in ((1.0, 1), (0.0, -1)):
        for i in range(n):
            for j in range(n):
                u, v = (i + 0.5) / n, (j + 0.5) / n
                p = [u, v]
                p.insert(axis, side)
                flux += normal * F(p)[axis] / (n * n)
print(f"flux through the unit cube = {flux:.6f}; triple integral of div F (= 3) over the cube = 3")

grad phi at (1, 2, 1): [4.0, 2.0, 6.0]
directional derivative = 6.666667 (20/3 = 6.666667)
div F at (1,1,1) = 5  curl F = (-1, -1, -1)
flux through the unit cube = 3.000000; triple integral of div F (= 3) over the cube = 3

Practice questions