MAST20029 · Engineering Mathematics
Vector Fields, Divergence and Curl
This opening chapter sets up the language of the whole vector-calculus block: a vector field F = F₁ i + F₂ j + F₃ k, the del operator ∇, and the three operations built from it — divergence (a scalar measuring net outflow), curl (a vector measuring local rotation) and gradient. It is examined mostly as a setup or first part of a larger Q1–Q3 vector-calculus question (compute a div/curl, or test conservativity), and the two identities curl(grad φ)=0 and div(curl F)=0 are reliable short-answer marks.
What this chapter covers
- 011. Vector fields F = F₁ i + F₂ j + F₃ k and how to picture them as an arrow at each point
- 022. The del operator ∇ = (∂/∂x, ∂/∂y, ∂/∂z) as the engine behind div, curl and grad
- 033. Divergence ∇·F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z — a SCALAR (net source/sink)
- 044. Curl ∇×F as the i,j,k determinant — a VECTOR (local rotation)
- 055. Gradient ∇φ of a scalar field, and that it points across level surfaces
- 066. Identity curl(grad φ) = 0 — every gradient field is curl-free
- 077. Identity div(curl F) = 0 — every curl is divergence-free
- 088. Reading the sign: positive divergence = source, circulating arrows = nonzero curl
Divergence and curl of a vector field, then a quick identity check
- 2 marks(a) Divergence is the sum of the matching partials: ∇·F = ∂(x²y)/∂x + ∂(yz²)/∂y + ∂(xz)/∂z = 2xy + z² + x.
- 3 marks(b) Curl is the i,j,k determinant ∇×F = (∂F₃/∂y − ∂F₂/∂z) i − (∂F₃/∂x − ∂F₁/∂z) j + (∂F₂/∂x − ∂F₁/∂y) k. Component by component: i: ∂(xz)/∂y − ∂(yz²)/∂z = 0 − 2yz = −2yz; j: −(∂(xz)/∂x − ∂(x²y)/∂z) = −(z − 0) = −z; k: ∂(yz²)/∂x − ∂(x²y)/∂y = 0 − x² = −x².
- 1 mark(c) div(curl F) = ∂(−2yz)/∂x + ∂(−z)/∂y + ∂(−x²)/∂z = 0 + 0 + 0 = 0, confirming the identity div(curl F) = 0.
Key terms
- Vector field
- A rule F(x,y,z) = F₁ i + F₂ j + F₃ k assigning a vector to each point of space — think of an arrow at every point (a velocity or force field).
- Del operator ∇
- The symbolic vector ∇ = (∂/∂x, ∂/∂y, ∂/∂z). Dotting it with F gives divergence, crossing it with F gives curl, and applying it to a scalar gives the gradient.
- Divergence ∇·F
- A SCALAR field, ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z, measuring the net rate at which the field flows out of a point — positive = source, negative = sink.
- Curl ∇×F
- A VECTOR field computed from the i,j,k determinant, measuring the local rotation (circulation per unit area) of F. Zero curl on a simply connected domain signals a conservative field.
- Gradient ∇φ
- For a scalar field φ, the vector (φ_x, φ_y, φ_z) that points in the direction of steepest increase and is normal to the level surfaces of φ.
- Vector identities
- Two facts used as instant checks: curl(grad φ) = 0 (every gradient field is irrotational) and div(curl F) = 0 (every curl is divergence-free).
Vector Fields, Divergence and Curl FAQ
What is the difference between divergence and curl?
Divergence is a scalar that measures how much the field spreads out from (or into) a point — like the net flow out of a tiny box. Curl is a vector that measures how much the field circulates around a point — like the rotation a tiny paddle wheel would feel. A field can have one without the other: a pure source has divergence but no curl, while a pure swirl has curl but no divergence.
How do I remember the curl determinant?
Write a 3×3 determinant with the top row i, j, k, the middle row the operators ∂/∂x, ∂/∂y, ∂/∂z, and the bottom row F₁, F₂, F₃, then expand along the top row. The crucial detail is the cofactor sign pattern + − +, so the j component picks up an extra minus. Practise the expansion until it is automatic.
Why are the identities curl(grad φ)=0 and div(curl F)=0 worth knowing?
They appear both as direct short-answer asks and as shortcuts. If a field is a gradient (F = ∇φ) you immediately know its curl is zero (so it is conservative); if you are asked to verify a field is a curl of something, its divergence must be zero. They also let you sanity-check a div/curl calculation.
How is this chapter examined?
Usually as the first part of a Q1–Q3 vector-calculus question: compute a divergence or curl, decide whether a field is conservative, or verify an identity. These are reliable, low-risk marks — the algebra is short, so the examiner rewards correct setup, the right determinant sign pattern and clear notation.
Exam move
Make the three operations reflexive: divergence = dot (scalar out), curl = cross (vector out, via the determinant with the + − + sign pattern), gradient = ∇ on a scalar. Drill a handful of fields until you can write div and curl without re-deriving the formula, and always pause to confirm the j-component minus sign in the curl. Memorise the two identities as both facts and checks — if you compute a curl, its divergence should come out zero. Because these parts are short and reliable, treat them as guaranteed marks: write the operation by name, show each partial derivative, and keep scalars and vectors visually distinct.
