Curl: definition, physical meaning, computation

At a Glance

• Frequency: 4 sub-parts across 3 of 13 years (2015, 2017, 2020, 2022)
• Priority tier: T3
• Marks (count): 10 (3), 12 (1)
• Average solve time: ~6 min
• Difficulty mix: easy 4
• Section: B | Dominant type: computation

Why This Chapter Matters

All four appearances of this atom are classified easy — this is guaranteed-mark territory. Every question follows an identical two-step pattern: set each component of $\nabla \times \vec F$ to zero to find unknown parameters, then integrate to recover the scalar potential. The 2017 question adds the minor twist of expressing the divergence in cylindrical coordinates (but since divergence is a scalar invariant, the value is just the Cartesian answer). Investing 15 minutes to internalize the curl formula and the potential-recovery algorithm pays off on four separate years.

Minimum Theory

Curl. For $\vec F = P\hat i + Q\hat j + R\hat k$: $\nabla \times \vec F = (R_y - Q_z)\hat i - (R_x - P_z)\hat j + (Q_x - P_y)\hat k.$ Remembered as the $3\times3$ determinant with rows $(\hat i, \hat j, \hat k)$, $(\partial_x, \partial_y, \partial_z)$, $(P, Q, R)$.

Irrotational field. $\nabla \times \vec F = \vec 0$ (all three components vanish). This is the necessary and sufficient condition (on a simply connected domain) for the existence of a scalar potential $\phi$ with $\vec F = \nabla\phi$.

Potential recovery. Given $\nabla\phi = \vec F = (P,Q,R)$:

1. Integrate $\phi_x = P$ with respect to $x$ → $\phi = \int P\,dx + g(y,z)$.
2. Differentiate w.r.t. $y$ and match $Q$ → determine $g_y$, integrate → $g = \int(\cdots)\,dy + h(z)$.
3. Differentiate w.r.t. $z$ and match $R$ → determine $h'(z)$, integrate.

Question Archetypes

irrotational-potential (2 questions; 2015, 2022)

Recognition Cues

• “Show/verify that $\vec F$ is irrotational” — compute curl, show all components vanish.
• “Find $\phi$ such that $\vec F = \nabla\phi$” — integrate successively.

Solution Template

1. Compute curl components. Three partial-derivative differences: $(R_y-Q_z)$, $(P_z-R_x)$, $(Q_x-P_y)$. Show each is 0.
2. Integrate $\phi_x = P$ w.r.t. $x$: $\phi = \int P\,dx + g(y,z)$.
3. Match $\phi_y = Q$: differentiate the result w.r.t. $y$; solve for $g_y$; integrate to get $g$.
4. Match $\phi_z = R$: differentiate w.r.t. $z$; solve for $h'(z)$; integrate.
5. State $\phi$ with arbitrary constant $C$.

Worked Example(s)

2022 Paper 1, 2022-P1-Q5e (10 marks)

Show $\vec A = (6xy+z^3)\hat i + (3x^2-z)\hat j + (3xz^2-y)\hat k$ is irrotational; find $\phi$.

Curl:

• $\hat i$: $\partial_y(3xz^2-y) - \partial_z(3x^2-z) = -1-(-1) = 0$ ✓
• $\hat j$: $\partial_z(6xy+z^3) - \partial_x(3xz^2-y) = 3z^2 - 3z^2 = 0$ ✓
• $\hat k$: $\partial_x(3x^2-z) - \partial_y(6xy+z^3) = 6x - 6x = 0$ ✓

Potential:

• $\phi_x = 6xy+z^3 \Rightarrow \phi = 3x^2y + xz^3 + g(y,z)$.
• $\phi_y = 3x^2 + g_y = 3x^2-z \Rightarrow g_y = -z \Rightarrow g = -yz + h(z)$.
• $\phi_z = 3xz^2 - y + h'(z) = 3xz^2-y \Rightarrow h'(z) = 0 \Rightarrow h = C$.

$\boxed{\phi = 3x^2y + xz^3 - yz + C.}$

2015 Paper 1, 2015-P1-Q7c (12 marks)

Verify $\vec F = (x^2+xy^2)\hat i + (y^2+x^2y)\hat j$ is irrotational; find scalar potential.

2D field (no $z$-component, no $z$-dependence): only the $\hat k$ curl component matters.

$\hat k$: $\partial_x(y^2+x^2y) - \partial_y(x^2+xy^2) = 2xy - 2xy = 0$ ✓. Irrotational.

Potential:

• $\phi_x = x^2+xy^2 \Rightarrow \phi = x^3/3 + x^2y^2/2 + g(y)$.
• $\phi_y = x^2y + g'(y) = y^2+x^2y \Rightarrow g'(y) = y^2 \Rightarrow g = y^3/3 + C$.

$\boxed{\phi = \frac{x^3}{3} + \frac{x^2y^2}{2} + \frac{y^3}{3} + C.}$

Common Traps

• For a 2D field, the curl has only one non-zero component: the $\hat k$ term $Q_x - P_y$. Only this needs to be checked.
• The potential is defined up to an additive constant $C$ — always include it.
• When integrating $\phi_x = P$ in $x$, the “constant of integration” is a function $g(y,z)$, not a scalar.

find-params-irrotational (2 questions; 2017, 2020)

Recognition Cues

• “For what values of $a,b,c$ is $\vec V$ irrotational?”
• The field has unknown parameters; setting curl components to zero gives 2–3 linear equations in $a,b,c$.

Solution Template

1. Write the three curl components as functions of $a,b,c$.
2. Set each to zero to get 3 equations.
3. Solve (usually one equation per parameter).
4. State the potential by integrating the field with the found parameters.

Worked Example(s)

2020 Paper 1, 2020-P1-Q5c (10 marks)

Find $a,b,c$ for $\vec V = (-4x-3y+az)\hat i + (bx+3y+5z)\hat j + (4x+cy+3z)\hat k$ to be irrotational.

Curl components:

• $\hat i$: $R_y - Q_z = c - 5 = 0 \Rightarrow c = 5$.
• $\hat j$: $P_z - R_x = a - 4 = 0 \Rightarrow a = 4$.
• $\hat k$: $Q_x - P_y = b - (-3) = b+3 = 0 \Rightarrow b = -3$.

$\boxed{a=4,\; b=-3,\; c=5.}$

Potential: with these values, integrate as in the template: $\phi = -2x^2 + \tfrac{3}{2}y^2 + \tfrac{3}{2}z^2 - 3xy + 4xz + 5yz + C.$

2017 Paper 1, 2017-P1-Q5d (10 marks)

Find $a,b,c$ for $\vec V = (x+y+az)\hat i + (bx+2y-z)\hat j + (-x+cy+2z)\hat k$ to be irrotational; find the divergence in cylindrical coordinates.

Curl:

• $\hat i$: $c - (-1) = c+1 = 0 \Rightarrow c = -1$.
• $\hat j$: $a - (-1) = a+1 = 0 \Rightarrow a = -1$.
• $\hat k$: $b - 1 = 0 \Rightarrow b = 1$.

$\boxed{a=-1,\; b=1,\; c=-1.}$

Divergence in cylindrical coordinates. The divergence $\nabla \cdot \vec V$ is a scalar invariant — its value is independent of the coordinate system. In Cartesian: $\nabla \cdot \vec V = \partial_x(x+y-z) + \partial_y(x+2y-z) + \partial_z(-x-y+2z) = 1+2+2 = 5.$ In cylindrical coordinates, the divergence formula gives the same constant $5$.

Common Traps

• Signs matter. The $\hat j$ component of curl is $P_z - R_x$ (not $R_x - P_z$). Missing a sign gives wrong parameters.
• Divergence is coordinate-invariant. The question asks for the divergence “in cylindrical coordinates” — but since the answer is a constant ($5$), no transformation is needed. State the invariance explicitly.
• Setting $Q_x - P_y = b + 3$ (since $P_y = -3$): a sign slip here gives $b = 3$ instead of $b = -3$.

Marks-Aware Writing

10-mark irrotational + potential: Show three curl components vanish (3 lines). Then three integration steps for the potential (3 lines). State the answer. Total ~7 lines.

10-mark find-params + divergence: Three one-line equations from curl = 0 (immediate). State the divergence (one line) and note coordinate invariance (one sentence).

Practice Set

• 2024-P1-Q7c (20 m) — curl and line integral;
• 2018-P1-Q8b (13 m) — irrotational field;
• 2017-P1-Q8c-ii (8 m) —