📐 Concept diagram

07-07 — Curl and Divergence

Phase: 7 — Calculus IV: Vector Calculus Subject: 07-07 Prerequisites: 07-06 — Green's Theorem, 07-04 — Vector Fields Next subject: 07-08 — Surface Integrals

Learning Objectives

By the end of this subject, you will be able to:

1. Compute the curl and divergence of 2D and 3D vector fields using the del operator ∇
2. Interpret curl as the infinitesimal circulation (rotation) and divergence as the infinitesimal expansion/compression of a vector field
3. Classify vector fields as irrotational (curl F = 0), solenoidal (div F = 0), or both
4. Apply vector identities involving curl, divergence, and gradient to simplify computations
5. Relate curl and divergence to physical concepts like fluid rotation, source/sink behavior, and incompressibility

Core Content

1. The Del Operator

⚠️ CRITICAL FOUNDATION: The del operator ∇ = ⟨∂/∂x, ∂/∂y, ∂/∂z⟩ produces curl (∇×F, measures rotation) and divergence (∇·F, measures expansion/compression). These two operations are the vocabulary of the fundamental theorems: Green's, Stokes', and Divergence.

The del (or nabla) operator ∇ is a vector differential operator:

$∇ = ⟨∂/∂x, ∂/∂y, ∂/∂z⟩
$

It is NOT a vector in the usual sense — it's an operator that acts on functions. But the algebraic notation is powerful.

Three ways to apply ∇:

1. Gradient (on a scalar f): ∇f = ⟨∂f/∂x, ∂f/∂y, ∂f/∂z⟩ — produces a vector field
2. Divergence (on a vector F): ∇ · F — produces a scalar field
3. Curl (on a vector F): ∇ × F — produces a vector field (3D) or scalar (2D)

2. Curl — The Rotation of a Vector Field

Definition (3D): For F(x, y, z) = ⟨P, Q, R⟩:

curl F = ∇ × F = |i    j    k   |
                 |∂/∂x ∂/∂y ∂/∂z|
                 |P    Q    R   |

= ⟨∂R/∂y − ∂Q/∂z,  ∂P/∂z − ∂R/∂x,  ∂Q/∂x − ∂P/∂y⟩

Physical interpretation: At a point, curl F points along the axis of local rotation, and its magnitude is the angular velocity of the rotation (specifically, curl F = 2ω where ω is the angular velocity vector of a small paddle wheel placed in the flow).

2D specialization: For F(x, y) = ⟨P, Q⟩, embed in 3D as ⟨P, Q, 0⟩:

curl F = ⟨0, 0, ∂Q/∂x − ∂P/∂y⟩

The scalar 2D curl is (curl F) · k = ∂Q/∂x − ∂P/∂y (exactly the integrand in Green's theorem).

Example 1: F(x, y, z) = ⟨xz, xyz, −y²⟩.

curl F = ⟨∂R/∂y − ∂Q/∂z, ∂P/∂z − ∂R/∂x, ∂Q/∂x − ∂P/∂y⟩
       = ⟨∂(−y²)/∂y − ∂(xyz)/∂z, ∂(xz)/∂z − ∂(−y²)/∂x, ∂(xyz)/∂x − ∂(xz)/∂y⟩
       = ⟨−2y − xy, x − 0, yz − 0⟩
       = ⟨−y(2+x), x, yz⟩

Example 2: F(x, y) = ⟨−y, x⟩ (rotational field).

curl F (2D scalar) = ∂Q/∂x − ∂P/∂y = 1 − (−1) = 2.

Positive curl → counterclockwise rotation. The field rotates with "strength" 2.

Example 3: F(x, y, z) = ⟨y, z, x⟩.

curl F = ⟨∂R/∂y−∂Q/∂z, ∂P/∂z−∂R/∂x, ∂Q/∂x−∂P/∂y⟩
       = ⟨0−1, 0−1, 0−1⟩ = ⟨−1, −1, −1⟩.

|curl F| = √3. The field has constant non-zero curl everywhere.

3. Divergence — Expansion and Compression

Definition: For F(x, y, z) = ⟨P, Q, R⟩:

$div F = ∇ · F = ∂P/∂x + ∂Q/∂y + ∂R/∂z
$

Physical interpretation: Divergence measures the net rate of flow out of (or into) a point per unit volume. Think of it as the "source strength":

• div F > 0: net outflow (source, expansion)
• div F < 0: net inflow (sink, compression)
• div F = 0: incompressible (solenoidal)

Example 4: F(x, y, z) = ⟨x, y, z⟩ (radial outward).

$div F = ∂x/∂x + ∂y/∂y + ∂z/∂z = 1 + 1 + 1 = 3.
$

Constant positive divergence — a "source" at every point.

Example 5: F(x, y, z) = ⟨−y, x, 0⟩ (rotation about z-axis).

$div F = 0 + 0 + 0 = 0.
$

The field is solenoidal (divergence-free) — it rotates without expanding or compressing.

Example 6: F(x, y, z) = ⟨x², y², z²⟩.

$div F = 2x + 2y + 2z = 2(x + y + z).
$

Divergence varies with position. In octants where x+y+z > 0, there's expansion; where x+y+z < 0, compression.

4. Classification of Vector Fields

5. Important Vector Identities

Identity 1: Curl of gradient is always zero.

$curl(∇f) = ∇ × (∇f) = 0
$

This means every gradient field is irrotational. The converse (on simply-connected domains): if curl F = 0, then F = ∇φ for some φ.

Proof sketch: ∇ × ∇f = ⟨f_zy − f_yz, f_xz − f_zx, f_yx − f_xy⟩ = ⟨0, 0, 0⟩ by Clairaut's theorem.

Identity 2: Divergence of curl is always zero.

$div(curl F) = ∇ · (∇ × F) = 0
$

This means every curl field is solenoidal. The converse (on the right kind of domain): if div F = 0, then F = curl G for some vector potential G.

Identity 3: Laplacian.

$div(∇f) = ∇ · ∇f = ∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z²
$

Identity 4: Curl of curl.

$curl(curl F) = ∇(div F) − ∇²F
$

Where ∇²F = ⟨∇²P, ∇²Q, ∇²R⟩ (component-wise Laplacian).

6. Physical Examples

Fluid velocity field v(x, y, z): - curl v is the vorticity — twice the angular velocity of fluid particles - div v is the dilation rate — rate of change of volume of a fluid element - For incompressible fluids: div v = 0 (conservation of mass) - For irrotational flows: curl v = 0 (potential flow)

Electromagnetic fields: - Electric field E: curl E = −∂B/∂t (Faraday's law), div E = ρ/ε₀ (Gauss's law) - Magnetic field B: curl B = μ₀J + μ₀ε₀ ∂E/∂t (Ampère-Maxwell), div B = 0 (no magnetic monopoles)

Gravitational field: g = ∇φ (conservative, irrotational). curl g = 0. div g = −4πGρ (Poisson's equation).

7. Geometric Interpretation

Curl: Place a tiny paddle wheel at a point. If the field causes it to rotate, curl ≠ 0. The direction of curl is along the axis of rotation (right-hand rule), and the magnitude is proportional to the angular speed.

Divergence: Draw a small box at a point. If more fluid flows out than in, div > 0. If flow is conserved (in = out), div = 0.

Common Misconceptions

1. "Curl means the field lines curve." A field can have curved field lines but zero curl (e.g., F = ⟨x, y⟩/r² — radial field). Curl measures local rotation, not curvature of field lines.

2. "Div F = 0 means the field doesn't spread out." It means local incompressibility. Field lines can still spread out — but the magnitude decreases to compensate (like the inverse-square field div(r̂/r²) = 0 away from origin).

3. "curl F = 0 implies F is the gradient of a potential." Only on simply-connected domains. On domains with holes, there may exist vector potentials instead (closed but not exact forms).

4. "Divergence and curl are independent." They are linked through identities: div(curl F) = 0 and curl(∇f) = 0. Together they form the Helmholtz decomposition: any (nice) vector field can be decomposed as F = −∇φ + curl A.

Key Terms

• Curl
• Divergence
• Gradient
• Harmonic
• Irrotational
• Rotational
• Solenoidal

Worked Examples

Example 1: Computing Both Operators

Problem: For F(x, y, z) = ⟨y²z, xz², x²y⟩, compute curl F and div F. Is F irrotational? Solenoidal?

Solution:

curl F = ⟨∂R/∂y − ∂Q/∂z, ∂P/∂z − ∂R/∂x, ∂Q/∂x − ∂P/∂y⟩

∂R/∂y = ∂(x²y)/∂y = x²
∂Q/∂z = ∂(xz²)/∂z = 2xz

∂P/∂z = ∂(y²z)/∂z = y²
∂R/∂x = ∂(x²y)/∂x = 2xy

∂Q/∂x = ∂(xz²)/∂x = z²
∂P/∂y = ∂(y²z)/∂y = 2yz

curl F = ⟨x² − 2xz, y² − 2xy, z² − 2yz⟩ ≠ 0. Not irrotational.

div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z = 0 + 0 + 0 = 0. Solenoidal! ✓

Example 2: Verifying curl(∇f) = 0

Problem: Let f(x, y, z) = x²y + yz³. Compute ∇f, then compute curl(∇f) and verify it's zero.

Solution:

$∇f = ⟨2xy, x² + z³, 3yz²⟩

Let F = ∇f = ⟨2xy, x²+z³, 3yz²⟩.

curl F = ⟨∂(3yz²)/∂y − ∂(x²+z³)/∂z, ∂(2xy)/∂z − ∂(3yz²)/∂x, ∂(x²+z³)/∂x − ∂(2xy)/∂y⟩
       = ⟨3z² − 3z², 0 − 0, 2x − 2x⟩
       = ⟨0, 0, 0⟩. ✓
$

Example 3: Finding a Vector Potential

Problem: Given F(x, y, z) = ⟨y, z, x⟩, we found curl F = ⟨−1, −1, −1⟩. Verify div(curl F) = 0.

Solution:

curl F = ⟨−1, −1, −1⟩.
div(curl F) = ∂(−1)/∂x + ∂(−1)/∂y + ∂(−1)/∂z = 0 + 0 + 0 = 0. ✓

Quiz

Q1: The curl of a vector field F = ⟨P, Q, R⟩ is defined as:

A) ∇ · F (divergence) B) ∇ × F (cross product of del with F) C) ∇f (gradient) D) ‖F‖ (magnitude)

Correct: B)

• If you chose B: curl F = ∇ × F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩. It measures the infinitesimal rotation of the field. Correct!
• If you chose A: That's divergence, a scalar measuring expansion/compression.
• If you chose C: Gradient applies to scalar functions, not vector fields.
• If you chose D: Magnitude says nothing about rotation.

Q2: The divergence of a vector field F = ⟨P, Q, R⟩ is:

A) ∇ × F B) ∇ · F = ∂P/∂x + ∂Q/∂y + ∂R/∂z C) ‖F‖ D) The gradient of F

Correct: B)

• If you chose B: div F = ∇ · F = ∂P/∂x + ∂Q/∂y + ∂R/∂z — a scalar measuring net outflow per unit volume. Correct!
• If you chose A: That's curl, a vector measuring rotation.
• If you chose C: Magnitude is not divergence.
• If you chose D: Gradient applies to scalar fields, not directly to vector fields.

Q3: A vector field with zero divergence everywhere is called:

A) Irrotational B) Solenoidal (incompressible) C) Conservative D) Harmonic

Correct: B)

• If you chose B: ∇ · F = 0 means the field is solenoidal — no sources or sinks, like an incompressible fluid. Correct!
• If you chose A: Irrotational means curl F = 0.
• If you chose C: Conservative means F = ∇f (curl F = 0 in simply connected regions).
• If you chose D: Harmonic applies to scalar functions with ∇²f = 0.

Q4: For any C² vector field F, the identity ∇ · (∇ × F) equals:

A) 0 (always) B) ∇ · F C) ‖F‖ D) 1

Correct: A)

• If you chose A: div(curl F) = 0 is a fundamental vector identity — the divergence of any curl is always zero. Correct!
• If you chose B: This would mean div(curl F) = div F, which is false in general.
• If you chose C: Not related to magnitude.
• If you chose D: div(curl F) is identically zero, not one.

Q5: Another fundamental identity: ∇ × (∇f) equals:

A) 0 (the zero vector) B) ∇ · f C) f D) ∇f

Correct: A)

• If you chose A: curl(grad f) = 0 — the curl of any gradient is the zero vector. Conservative fields are irrotational. Correct!
• If you chose B: ∇ · f is not defined for a scalar f.
• If you chose C: The curl of a gradient is zero, not the original function.
• If you chose D: The curl of a gradient is zero, not the gradient itself.

Q6: For F = ⟨−y, x, 0⟩ (a 2D rotational field), curl F equals:

A) ⟨0, 0, 0⟩ B) ⟨0, 0, 2⟩ C) 0 (scalar) D) ⟨0, 0, −2⟩

Correct: B)

• If you chose B: curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩ = ⟨0 − 0, 0 − 0, 1 − (−1)⟩ = ⟨0, 0, 2⟩. The field rotates counterclockwise with constant vorticity 2. Correct!
• If you chose A: The field clearly has rotation — curl is nonzero.
• If you chose C: Curl is a vector, not a scalar.
• If you chose D: Sign error — Q_x = 1, P_y = −1, so Q_x − P_y = 2.

Practice Problems

(Answers are below. Try each problem before checking.)

Problem 1: Compute curl F and div F for F(x, y, z) = ⟨x²z, xyz, yz²⟩.

Problem 2: Determine whether F(x, y) = ⟨x², −2xy⟩ is incompressible (div = 0). Also compute its 2D curl.

Problem 3: For F(x, y, z) = ⟨y, −x, 0⟩, compute curl F and interpret the result.

Problem 4: Verify that div(curl F) = 0 for F(x, y, z) = ⟨e^x sin y, e^x cos y, z⟩.

Problem 5: Classify each field as irrotational, solenoidal, both, or neither: (a) F = ⟨x, y, z⟩ (b) F = ⟨−y, x, 0⟩ (c) F = ⟨x, −y, 0⟩

Problem 6: For f(x, y, z) = xyz, compute ∇²f = div(∇f).

Problem 7: Show that the field F(x, y, z) = ⟨yz, xz, xy⟩ is the curl of some vector field G. Find G.

Summary

1. The curl ∇ × F measures local rotation of a vector field: its direction is the axis of rotation and its magnitude is the angular velocity; curl F = 0 characterizes irrotational (conservative) fields
2. The divergence ∇ · F measures local expansion/compression: positive divergence indicates sources (outflow), negative indicates sinks (inflow), and div F = 0 characterizes incompressible (solenoidal) fields
3. Two fundamental identities hold for any smooth scalar f and vector F: curl(∇f) = 0 (gradients are irrotational) and div(curl F) = 0 (curls are solenoidal)
4. The 2D curl ∂Q/∂x − ∂P/∂y is the scalar version appearing in Green's theorem; it represents circulation per unit area
5. Together, curl and divergence provide a complete local characterization of a vector field's behavior, and the Helmholtz decomposition theorem states that any (decaying) field can be uniquely decomposed into irrotational and solenoidal parts

Pitfalls

1. Confusing curl with curvature of field lines. A radial field like F = ⟨x, y⟩ has straight field lines but zero curl. A field with curved field lines can still be irrotational — curl measures local rotation (what a paddle wheel feels), not the visual curvature of the vector plot.

2. Assuming curl F = 0 always implies F = ∇φ. This is true only on simply-connected domains. The field F = ⟨−y, x⟩/(x² + y²) has zero curl away from the origin but is not a gradient field on any domain encircling the origin — evidenced by its nonzero circulation around the origin.

3. Thinking div F = 0 means the field doesn't spread out. Incompressibility means the field's magnitude adjusts to compensate for geometric spreading. The inverse-square field r̂/r² has zero divergence everywhere except at the origin, even though its field lines spread out radially — the magnitude decays as 1/r² to keep the flux through any sphere constant.

4. Misapplying the cyclic property when computing identities. While div(curl F) = 0 and curl(∇f) = 0 always hold, the converse — "if div F = 0 then F = curl G" — requires the domain to be contractible (no holes in 3D). Don't assume a vector potential exists on every domain.

5. Computing curl as a cross product without checking variable dependencies. Curl is ∇ × F, not a literal cross product of fixed vectors. Each component involves partial derivatives with respect to different variables — ∂/∂y of R, ∂/∂z of Q, etc. A common mistake is differentiating with respect to the wrong variable or forgetting which component maps to which term.

Next Steps

Move on to 07-08 — Surface Integrals to learn how to integrate scalar functions and vector fields over surfaces in 3D, compute surface area, and evaluate flux integrals — the bridge to Stokes' and the Divergence Theorems.