Lecture 3 Curl and Divergence

Lecture #3 — Curl and Divergence

Course: Calculus & Linear Algebra (TU Delft)
Reference: Stewart §16.5#

1. The Nabla Operator (∇)

The vector differential operator nabla (or del) is defined as:

\nabla = ⟨ \frac{\partial}{\partial x_{1}}, \frac{\partial}{\partial x_{2}}, \dots, \frac{\partial}{\partial x_{n}} ⟩

In $R^{3}$ this becomes:

\nabla = ⟨ \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} ⟩ = i \frac{\partial}{\partial x} + j \frac{\partial}{\partial y} + k \frac{\partial}{\partial z}

2. Gradient and Divergence

Definitions

Gradient of a scalar function $f : R^{n} \to R$ :

grad (f) = \nabla f = ⟨ \frac{\partial f}{\partial x_{1}}, \frac{\partial f}{\partial x_{2}}, \dots, \frac{\partial f}{\partial x_{n}} ⟩

Divergence of a vector field $F : R^{n} \to R^{n}$ :

div (F) = \nabla \cdot F = \frac{\partial F_{1}}{\partial x_{1}} + \frac{\partial F_{2}}{\partial x_{2}} + \dots + \frac{\partial F_{n}}{\partial x_{n}}

Key distinction: $\nabla f$ is a vector field, while $\nabla \cdot F$ is a scalar field.

In $R^{3}$ : $\nabla \cdot F = \frac{\partial F_{1}}{\partial x} + \frac{\partial F_{2}}{\partial y} + \frac{\partial F_{3}}{\partial z}$

3. Curl of a Vector Field

Definition

For a vector field $F : R^{3} \to R^{3}$ , the curl (or rotation) is:

curl (F) = \nabla \times F = ⟨ \frac{\partial F_{3}}{\partial y} - \frac{\partial F_{2}}{\partial z}, \frac{\partial F_{1}}{\partial z} - \frac{\partial F_{3}}{\partial x}, \frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y} ⟩

For 2D fields $F = ⟨ F_{1}, F_{2} ⟩$ in $R^{2}$ , extend to 3D as $\tilde{F} = ⟨ F_{1}, F_{2}, 0 ⟩$ . Then:

curl (F) = \nabla \times \tilde{F} = ⟨ 0, 0, \frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y} ⟩

Note: $curl (F)$ is always a vector field in $R^{3}$ .

4. Physical Interpretation

Divergence

The divergence measures net outward flux per unit volume at a point:

\nabla \cdot F (P) = lim_{ϵ \to 0^{+}} \frac{1}{Vol (B_{ϵ})} \oiint_{\partial B_{ϵ}} F \cdot d S

$\nabla \cdot F (P) > 0$ : point $P$ is a source (field flows outward)
$\nabla \cdot F (P) < 0$ : point $P$ is a sink (field flows inward)

Curl

The curl measures circulation per unit area at a point. Projecting onto a direction $\hat{n}$ gives:

(\nabla \times F (P)) \cdot \hat{n} = lim_{ϵ \to 0^{+}} \frac{1}{Area (S_{\hat{n}} (ϵ))} \oint_{\partial S_{\hat{n}}} F \cdot d r

The direction of $curl (F)$ is given by the right-hand rule: curl your right hand's fingers in the rotation direction — your thumb points in the direction of $curl (F)$ .

5. Application: Maxwell's Equations

Equation	Name
$\nabla \cdot E = \frac{ρ}{ε_{0}}$	Gauss' law for electricity
$\nabla \cdot B = 0$	Gauss' law for magnetism
$\nabla \times E = - \frac{\partial B}{\partial t}$	Faraday's law
$\nabla \times B = μ_{0} (J + ε_{0} \frac{\partial E}{\partial t})$	Ampère's law

Physical insight: In a vacuum ( $ρ = 0$ , $J = 0$ ), using the identity $μ_{0} ε_{0} = \frac{1}{c^{2}}$ , Maxwell's equations yield the wave equation:

\frac{\partial^{2} E}{\partial t^{2}} = c^{2} \nabla^{2} E

This shows that light is an electromagnetic wave.

Note: Maxwell's equations do not need to be memorized for this course.

6. Second-Order Derivatives

The Laplacian

The Laplace operator (Laplacian) is:

\nabla^{2} = \nabla \cdot \nabla = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}

It appears in many fundamental PDEs: the diffusion equation $\frac{\partial T}{\partial t} = α \nabla^{2} T$ , the wave equation $\frac{\partial^{2} u}{\partial t^{2}} = v^{2} \nabla^{2} u$ , and Laplace's equation $\nabla^{2} u = 0$ .

Key Identities

For a smooth scalar field $f$ and vector field $F$ :

Identity	Meaning
$\nabla \cdot (\nabla f) = \nabla^{2} f$	Divergence of a gradient is the Laplacian
$\nabla \cdot (\nabla \times F) = 0$	Curl is always solenoidal
$\nabla \times (\nabla f) = 0$	Gradient is always irrotational
$\nabla \times (\nabla \times F) = \nabla (\nabla \cdot F) - \nabla^{2} F$	Vector Laplacian identity

These identities don't need to be memorized, but you should be able to apply them.

7. Irrotational and Solenoidal Vector Fields

Definitions

A vector field $F$ is irrotational if $\nabla \times F = 0$
A vector field $F$ is solenoidal if $\nabla \cdot F = 0$

Irrotational Fields and Conservative Fields

Every conservative field is irrotational:

F = \nabla ϕ ⟹ \nabla \times F = 0

On an open, simply-connected region $D \subset R^{3}$ , the converse also holds:

F = \nabla ϕ ⟺ \nabla \times F = 0

Watch out: An irrotational field on a non-simply-connected domain (e.g., $R^{3}$ minus a line) is not necessarily conservative. In such cases the field is irrotational but not conservative.

Solenoidal Fields and Vector Potentials

The curl of any vector field is solenoidal:

F = \nabla \times G ⟹ \nabla \cdot F = 0

The field $G$ is called a vector potential for $F$ . The magnetic field $B$ (with $\nabla \cdot B = 0$ ) is the classical example.

8. Summary Table

Property	Condition	Consequence
Conservative	$F = \nabla ϕ$	Path-independent line integrals
Irrotational	$\nabla \times F = 0$	No local rotation/circulation
Solenoidal	$\nabla \cdot F = 0$	No net outward flux (no sources/sinks)

9. Practice Problems

Exercise 1 — Compute div and curl

(a) $F (x, y, z) = ⟨ 3 x, - 2 y, x^{2} + z ⟩$ → $div (F) = 2$ , $curl (F) = ⟨ 0, - 2 x, 0 ⟩$

(b) $F (x, y) = ⟨ \sin (x), e^{y} - x ⟩$ → $div (F) = \cos (x) + e^{y}$ , $curl (F) = ⟨ 0, 0, - 1 ⟩$

Exercise 2 — Irrotational or Solenoidal?

(a) $F = ⟨ x, y, z ⟩$ : Irrotational ✓ (curl = 0)

(b) $F = ⟨ y z, x^{2} y, 3 z ⟩$ : Not irrotational ✗

(d) $F = ⟨ x^{2}, 3 x y, - x^{2} z - 3 x y z ⟩$ : Not solenoidal ✗

Exercise 3 — Maxwell's equations

If $E (x, y, z) = ⟨ 3 x^{2}, 6 x y - y, x y z^{2} ⟩$ , what is the charge density $ρ$ at $(0, 1, 2)$ ?

\nabla \cdot E = 6 x + (6 x - 1) + 2 x y z = \frac{ρ}{ε_{0}}

At $(0, 1, 2)$ : $\nabla \cdot E = 0 + (- 1) + 0 = - 1$ , so $ρ = - ε_{0}$

10. What's Next

Lecture 4: Stokes' theorem (Stewart §16.8)

Stokes' theorem relates the flux of $curl (F)$ through a surface to the line integral of $F$ around its boundary — a beautiful generalization connecting curl, surfaces, and line integrals.