Vector Calculus: grad, div, curl, and Integrals

The classical vector-analysis layer (analysis spine). We introduce scalar and vector fields, the three differential operators gradient, divergence, and curl, and the two new kinds of integral — line integrals and surface integrals — that the integral theorems relate. This is the concrete $R^{3}$ language that integral-theorems.md proves theorems in, and that differential-forms.md later unifies.

References: Spivak, Calculus on Manifolds; Marsden & Tromba, Vector Calculus.

1. Scalar and Vector Fields

On an open set $U \subseteq R^{3}$ :

a scalar field is a function $f : U \to R$ ;
a vector field is a map $F : U \to R^{3}$ , $F = (F_{1}, F_{2}, F_{3})$ .

We assume fields are as smooth as needed (usually $C^{1}$ or $C^{2}$ ). Write the del operator formally as $\nabla = (\partial_{x}, \partial_{y}, \partial_{z})$ .

2. The Three Differential Operators

Operator	Input → output	Formula
Gradient	scalar → vector	$\nabla f = (\partial_{x} f, \partial_{y} f, \partial_{z} f)$
Divergence	vector → scalar	$\nabla \cdot F = \partial_{x} F_{1} + \partial_{y} F_{2} + \partial_{z} F_{3}$
Curl	vector → vector	$\nabla \times F = (\partial_{y} F_{3} - \partial_{z} F_{2}, \partial_{z} F_{1} - \partial_{x} F_{3}, \partial_{x} F_{2} - \partial_{y} F_{1})$

The Laplacian is $Δ f = \nabla \cdot \nabla f = \partial_{x}^{2} f + \partial_{y}^{2} f + \partial_{z}^{2} f$ . Interpretations: $\nabla f$ points uphill (multivariable.md §2.2); $\nabla \cdot F$ measures the net outflow per unit volume (source/sink density); $\nabla \times F$ measures local circulation (rotation) of the field.

2.1 The two fundamental identities

For $C^{2}$ fields, $\nabla \times (\nabla f) = 0, \nabla \cdot (\nabla \times F) = 0.$

Both are immediate from the symmetry of mixed partials (multivariable.md §4): each component is a difference like $\partial_{y} \partial_{z} f - \partial_{z} \partial_{y} f = 0$ . These two identities are the same fact $d^{2} = 0$ written for $0$ -forms and $1$ -forms (differential-forms.md §4), and they constrain which fields can be gradients or curls (§3, §5).

2.2 Product identities

Useful in proofs (e.g. Green's identities, the divergence theorem applied to products):

$\nabla \cdot (f F) = \nabla f \cdot F + f \nabla \cdot F, \nabla \times (f F) = \nabla f \times F + f \nabla \times F .$

3. Line Integrals

Let $γ : [a, b] \to U$ be a piecewise- $C^{1}$ curve.

Of a scalar field (arc length): $\int_{γ} f d s = \int_{a}^{b} f (γ (t)) ∥ γ^{'} (t) ∥ d t$ .
Of a vector field (work/circulation): $\int_{γ} F \cdot d r = \int_{a}^{b} F (γ (t)) \cdot γ^{'} (t) d t$ .

The vector line integral is independent of the parametrization (up to orientation; reversing $γ$ flips the sign). A closed curve's integral $\oint_{γ} F \cdot d r$ is the circulation.

3.1 Conservative fields and path independence

$F$ is conservative on $U$ if $F = \nabla f$ for some potential $f$ . Then the line integral is path-independent:

Gradient theorem (FTC for line integrals). If $F = \nabla f$ then $\int_{γ} F \cdot d r = f (γ (b)) - f (γ (a)) .$

Proof. The integrand is $\frac{d}{d t} f (γ (t))$ by the chain rule (multivariable.md §3); apply the one-variable FTC (riemann-integral.md §5). $□$

This is itself a baby Stokes theorem (the $1$ -dimensional case). Consequences: $\oint F \cdot d r = 0$ for every loop iff $F$ is conservative. A conservative field is necessarily irrotational ( $\nabla \times F = 0$ , by §2.1). The converse holds on simply connected domains (metric-spaces.md §6) but fails otherwise — the field $F = (- y, x, 0) / (x^{2} + y^{2})$ is irrotational on $R^{3}$ minus the $z$ -axis yet has nonzero circulation around it. This gap is measured by de Rham cohomology (differential-forms.md §6).

4. Surface Integrals

Let $S$ be a surface parametrized by $r : D \to R^{3}$ , $D \subseteq R^{2}$ , with partials $r_{u}, r_{v}$ . The existence of such parametrizations for regular level surfaces is guaranteed by the implicit function theorem (inverse-implicit.md §3).

Surface area element: $d S = ∥ r_{u} \times r_{v} ∥ d u d v$ .
Of a scalar field: $\iint_{S} f d S = \iint_{D} f (r) ∥ r_{u} \times r_{v} ∥ d u d v$ .
Orientation and the normal: a choice of unit normal $n = \frac{r _{u} \times r _{v}}{∥ r _{u} \times r _{v} ∥}$ orients $S$ . (Non-orientable surfaces like the Möbius band admit no consistent choice.)
Flux of a vector field: $\iint_{S} F \cdot d S = \iint_{S} F \cdot n d S = \iint_{D} F (r) \cdot (r_{u} \times r_{v}) d u d v .$

Flux measures the net rate at which $F$ crosses $S$ . Reversing orientation flips its sign. The well-definedness (independence of parametrization, up to orientation) follows from the change-of-variables formula (multiple-integrals.md §4) applied to reparametrizations of $D$ .

5. The Operator → Theorem Dictionary

Each differential operator pairs with an integral theorem, all instances of one pattern ("integrate the derivative over a region = integrate the field over its boundary"):

Operator	Theorem	Statement
$\nabla f$	Gradient theorem (§3.1)	$\int_{γ} \nabla f \cdot d r = f ∣_{\partial γ}$
$\nabla \times F$	Stokes' theorem	$\iint_{S} (\nabla \times F) \cdot d S = \oint_{\partial S} F \cdot d r$
$\nabla \cdot F$	Divergence theorem	$∭_{V} (\nabla \cdot F) d V = \iint_{\partial V} F \cdot d S$

The middle and bottom rows are proved in integral-theorems.md; the unifying pattern is made precise in stokes-general.md.

6. Where this page is used

The operators and integrals defined here are the subject of the classical Green's, Stokes', and divergence theorems (integral-theorems.md).
The identities $\nabla \times \nabla = 0$ , $\nabla \cdot (\nabla \times) = 0$ become $d^{2} = 0$ , and grad/curl/div become the exterior derivative in dimensions $0, 1, 2$ (differential-forms.md).
Flux and circulation are the physical readings of $F^{μν}$ in covariant electromagnetism (SR/fields/electromagnetism.md).

Next: The Classical Integral Theorems.

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