Vector calculus

The Helmholtz decomposition separates a vector field into a gradient field and a field with zero divergence. The precise function spaces, boundary conditions and decay hypotheses determine existence and uniqueness. In the square-integrable bounded-domain formulation, the solenoidal component is selected by the Leray-Helmholtz projection.

Vector field

 2  0

A vector field assigns a vector to every point of its domain.

Time-dependent vector field ( $\partial_{t} ρ_{t} = Y_{t} \circ ρ_{t}$ )

A smooth family

Y_{t}

assigns a tangent vector at each space-time point. Its flow maps solve

\partial_{t} ρ_{t} = Y_{t} \circ ρ_{t}

, with

ρ_{0}

the identity. For a varying differential form, differentiation gives

\partial_{t} (ρ_{t}^{*} α_{t}) = ρ_{t}^{*} (\overset{α}{˙}_{t} + L_{Y_{t}} α_{t})

Finite-time flow completeness on a compact manifold

A smooth time-dependent vector field on a compact manifold without boundary has flow maps throughout each compact time interval on which the field is defined. Finite-time escape is impossible on the compact state space. Uniqueness and the backward-time equation supply smooth inverses, so the flow is a smooth isotopy.

Radial dilation tangency to homogeneous zero sets ( $H (λ x) = λ^{d} H (x) ⟹ x \cdot \nabla H = d H$ )

Differentiate the displayed homogeneity relation in

λ

1

. On every regular point of

H = 0

, the radial vector field

x

is therefore perpendicular to the normal gradient and tangent to the level surface. Its flow

x (t) = e^{t} x (0)

preserves that zero set. At a singular vertex this flow formulation remains meaningful even if there is no tangent plane.

Divergence and curl of a radial vector field ( $\nabla \cdot (f (r) x) = r f^{'} (r) + n f (r), \nabla \times (f (r) x) = 0 (n = 3)$ )

For

r = ∣ x ∣ > 0

, the gradient satisfies

\partial_{i} r = x_{i} / r

. The product rule and Einstein notation then give the displayed divergence formula in dimension

n

. In dimension three, its derivative matrix is symmetric, so the curl vanishes. Zero divergence forces

f (r) = C r^{- n}

on each positive radial interval. This formula does not assert regularity at the origin.

Azimuthal inverse-radius vector field ( $G = e_{ϕ} / ρ$ )

The vector field

G = (- y, x, 0) / (x^{2} + y^{2})

has zero curl away from its excluded axis, but has circulation

2 π

around a positively oriented circle enclosing that axis. Stokes theorem gives zero total circulation for a smooth annular surface avoiding the axis, whose inner and outer boundary circles have opposite orientations. It cannot be applied directly to a disk meeting the axis, because the field is undefined there.

Integral curve of a vector field ( $\overset{γ}{˙} (s) = X (γ (s))$ )

An integral curve follows a vector field by solving the displayed ordinary differential equation. For a smooth vector field, integral curves through initial points assemble into its local flow.

Curvature of an integral curve of a vector field ( $κ = ∣ F \times (F \cdot \nabla) F ∣/∣ F ∣^{3}$ )

For a continuously differentiable nonzero vector field

F

, the curvature of a space curve following its direction is

κ = \frac{∣ F \times ( F \cdot \nabla ) F ∣}{∣ F ∣ ^{3}} .

Take the unit tangent vector

t = \pm F /∣ F ∣

and differentiate with respect to arc length. The derivative of

∣ F ∣^{- 1}

is parallel to

F

and disappears from

t \times t^{'}

. Since

t \cdot t^{'} = 0

, we have

∣ t \times t^{'} ∣ = ∣ t^{'} ∣ = κ

, proving the formula. It remains valid with zero curvature, even where the usual Frenet frame is undefined.

Axisymmetric vector field

An axisymmetric vector field is invariant under rotations about a chosen axis. In cylindrical coordinates, its components are independent of the azimuthal angle.

Solenoidal vector field

A solenoidal vector field has zero divergence. Magnetic fields are solenoidal because

\nabla \cdot B = 0

Spherically symmetric function

A scalar function on Euclidean space is spherically symmetric when it depends only on the distance

r = ∣ x ∣

from a fixed center.

Helicity conservation by tangent boundary conditions

If a helicity density evolves as a sum of directional derivatives along divergence-free fields tangent to the boundary, its volume integral is conserved because both terms become vanishing boundary fluxes.

Divergence ( $\nabla \cdot$ )

For

F = (F_{1}, F_{2}, F_{3})

in Cartesian coordinates,

\nabla \cdot F = \partial_{i} F_{i} = \frac{\partial F _{1}}{\partial x} + \frac{\partial F _{2}}{\partial y} + \frac{\partial F _{3}}{\partial z} .

Divergence of a Riemannian vector field ( $div_{g} X = \nabla_{i} X^{i}$ )

For the Levi-Civita connection of a Riemannian metric, the divergence of

X

div_{g} X = tr (Y \mapsto \nabla_{Y} X)

. In coordinates it equals

(det g)^{- 1/2} \partial_{i} ((det g)^{1/2} X^{i})

. On an oriented manifold it satisfies

L_{X} ω_{g} = (div_{g} X) ω_{g}

. The codifferential on a one-form satisfies

δ θ = - div_{g} (θ^{♯})

, where

♯

is the musical isomorphism; this follows by integration by parts or the Hodge star formula.

Cylindrically radial divergence ( $\nabla \cdot [f (ρ) e_{ρ}] = ρ^{- 1} (ρ f)'$ )

For a vector field directed radially from a fixed axis and depending only on distance

ρ

from that axis, cylindrical coordinates give

\nabla \cdot [f (ρ) e_{ρ}] = f' + f / ρ

. Zero divergence gives

f = A / ρ

on any positive radial interval. Unless

A = 0

, this field is singular on the axis, so applications of the divergence theorem must use a domain avoiding it or account separately for that singularity.

Distributional divergence ( $⟨ div z, η ⟩ = - \int z \cdot \nabla η$ )

For a locally integrable vector field, this formula defines its divergence as a distribution. For a piecewise smooth field, a jump of its normal component across an interface adds a surface measure to the smooth divergences. A continuous normal component eliminates that measure, as follows by integration by parts on the two sides.

Divergence in polar coordinates ( $\nabla \cdot u = \frac{1}{r} \partial_{r} (r u_{r}) + \frac{1}{r} \partial_{θ} u_{θ}$ )

For

u = u_{r} e_{r} + u_{θ} e_{θ}

\nabla \cdot u = \frac{1}{r} \frac{\partial ( r u _{r} )}{\partial r} + \frac{1}{r} \frac{\partial u _{θ}}{\partial θ} .

Product rule for divergence

For a scalar field

ϕ

and vector field

F

\nabla \cdot (ϕF) = \nabla ϕ \cdot F + ϕ \nabla \cdot F .

Total divergence

A total divergence has the form

\nabla \cdot F

. The divergence theorem converts its volume integral into a boundary flux, which vanishes when the field has suitable decay or boundary conditions.

Divergence of a curl is zero ( $\nabla \cdot (\nabla \times F) = 0$ )

For a twice continuously differentiable vector field, commuting partial derivatives and antisymmetry of the Levi-Civita symbol give

\nabla \cdot (\nabla \times F) = 0.

Curl ( $\nabla \times$ )

The curl of

F

has components

(\nabla \times F)_{i} = ϵ_{ijk} \partial_{j} F_{k} .

Irrotational vector field ( $\nabla \times X = 0$ )

A continuously differentiable vector field on an open subset of three-dimensional Euclidean space is irrotational when its curl is zero. Under the Euclidean identification

α = X_{i} d x^{i}

, this says that the differential one-form

α

is a closed differential form. The Poincare lemma gives a potential of a conservative vector field locally. A global potential additionally requires zero line integrals around every closed curve; simply connected domains ensures this. On the punctured plane extended in the third direction,

X = (- y, x, 0) / (x^{2} + y^{2})

has zero curl but integral

2 π

around a circle, so it is locally conservative and has no single-valued global potential.

Vector potential

A vector potential for a divergence-free vector field

F

is a vector field

A

satisfying

F = \nabla \times A

. It is not unique because adding a gradient leaves its curl unchanged.

Radial vector potential of a solenoidal vector field ( $A (x) = D (x) \times x$ )

Let

B

be a continuously differentiable solenoidal vector field on an open star-shaped set containing the origin. Then

A (x) = (\int_{0}^{1} tB (t x) d t) \times x

is a vector potential for

B

. For

D = \int_{0}^{1} tB (t x) d t

, differentiation gives

\nabla \cdot D = 0

and

\nabla \times (D \times x) = 2 D + (x \cdot \nabla) D = \int_{0}^{1} \partial_{t} (t^{2} B (t x)) d t = B (x)

. Boundedness near the origin removes the lower endpoint. A field defined only on a punctured domain need not satisfy these hypotheses.

Curl of the curl identity

For a twice differentiable vector field

A

\nabla \times (\nabla \times A) = \nabla (\nabla \cdot A) - \nabla^{2} A .

Laplacian ( $Δ$ )

The Laplacian of a twice differentiable scalar field is the divergence of its gradient,

\nabla^{2} f = \nabla \cdot \nabla f

; it acts componentwise on vector fields.

Laplacian in polar coordinates ( $Δ = \frac{1}{r} \partial_{r} (r \partial_{r}) + \frac{1}{r ^{2}} \partial_{θ}^{2}$ )

For a twice differentiable scalar field on the plane,

\nabla^{2} f = \frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial f}{\partial r}) + \frac{1}{r ^{2}} \frac{\partial ^{2} f}{\partial θ ^{2}} .

Product rule for the Laplacian

For twice differentiable scalar fields,

\nabla^{2} (f g) = f \nabla^{2} g + 2\nabla f \cdot \nabla g + g \nabla^{2} f .

Biharmonic operator ( $Δ^{2}$ )

The biharmonic operator applies the Laplacian twice. Under clamped boundary conditions, two integrations by parts give the positive quadratic form

⟨ Δ^{2} u, u ⟩ = \int_{Ω} ∣Δ u ∣^{2} .

Clamped biharmonic problem ( $Δ^{2} u = f, u = \partial_{n} u = 0$ )

The clamped problem uses the weak solution identity

\int Δ u Δ v = \int f v

for every

v

in the clamped second-order Sobolev space. For

f \in L^{2} (U)

, the clamped Hessian identity and the Poincare inequality make this a bounded coercive bilinear form. The Lax-Milgram theorem gives a unique weak solution without a mean-zero condition on

f

Biharmonic equation ( $\nabla^{4} f = 0$ )

The biharmonic equation is

\nabla^{4} f = 0

, where the biharmonic operator is the square of the Laplacian. In planar Stokes flow, taking the curl of the momentum equation and using a Cartesian streamfunction produces this equation. Its decaying spatial Fourier series modes have form

(A + B y) e^{- ∣ k ∣ y} e^{ik x}

Divergence and curl of a cross product

For smooth vector fields

F, G

\nabla \cdot (F \times G) = G \cdot (\nabla \times F) - F \cdot (\nabla \times G)

and

\nabla \times (F \times G) = F (\nabla \cdot G) - G (\nabla \cdot F) + (G \cdot \nabla) F - (F \cdot \nabla) G .

(2)

Both follow by contracting two Levi-Civita symbols and applying the product rule.

Curl of a cross product ( $\nabla \times (U \times v) = U \nabla \cdot v - v \nabla \cdot U + (v \cdot \nabla) U - (U \cdot \nabla) v$ )

The displayed vector calculus identity follows by contracting the two alternating tensors in the component formula for curl. If both fields have zero divergence, only the two directional derivatives remain. It rewrites the transport terms in the adjoint linearized Navier-Stokes evolution.

Levi-Civita symbol

 2  0

In three dimensions,

ϵ_{ijk}

is zero when indices repeat and is the sign of the permutation

(i, j, k)

otherwise.

Rotation invariance of the Levi-Civita symbol ( $R_{ia} R_{jb} R_{k c} ϵ_{ab c} = (det R) ϵ_{ijk}$ )

The determinant transformation formula for the Levi-Civita symbol shows that it is invariant under every proper rotation matrix, for which

det R = 1

. Under an orthogonal reflection its sign reverses. This distinguishes proper-rotation isotropy from the pseudotensor transformation convention when both orientations are included.

Contraction of two Levi-Civita symbols

Contracting one index gives

ϵ_{ijk} ϵ_{i pq} = δ_{j p} δ_{k q} - δ_{j q} δ_{k p} .

Lagrange identity for the cross product

For three-dimensional vectors,

∣ x \times y ∣^{2} = ∣ x ∣^{2} ∣ y ∣^{2} - (x \cdot y)^{2} .

The corresponding identity in

R^{n}

∣ x ∣^{2} ∣ y ∣^{2} - (x \cdot y)^{2} = \frac{1}{2} \sum_{i, j} (x_{i} y_{j} - x_{j} y_{i})^{2} .

(2)

Vector triple product

Contracting two Levi-Civita symbols gives

a \times (b \times c) = b (a \cdot c) - c (a \cdot b) .

Feasible inner products with two unit vectors

For linearly independent unit vectors

y, z

with

c = y \cdot z

, a pair

p = x \cdot y, q = x \cdot z

is attained by some unit vector

x

exactly when

\frac{p ^{2} - 2 c pq + q ^{2}}{1 - c ^{2}} \leq 1.

(2)

The left-hand side is the squared norm of the least-norm vector having those two inner products.

Conservative vector field

A vector field

X

is conservative when

X = \nablaΦ

for a potential of a conservative vector field

Φ

. On a simply connected open subset of

R^{3}

, every continuously differentiable curl-free vector field is conservative. Locally this follows from the Poincare lemma applied to the differential one-form

X_{i} d x^{i}

; globally, simple connectivity makes its line integral independent of the path. The fundamental theorem for line integrals expresses that path independence as the difference of the potential at the endpoints.

Periods obstruct a periodic potential

If a periodic vector field has a periodic potential of a conservative vector field, the line integral between points separated by a period vector must vanish, since it is the potential difference. A periodic gradient field need not satisfy this: the constant field

(1, 2)

on the plane has potential

x + 2 y

, but its integrals over the two unit translations are

1

and

2

. On the periodic quotient these translation integrals are nonzero periods, obstructing a single-valued potential there.

Potential of a conservative vector field ( $X = \nablaΦ$ )

A potential for a conservative vector field

X

is a function with scalar values

Φ

whose gradient is

X

. Potentials differ by a constant on each connected component. The fundamental theorem for line integrals gives

\int_{γ} X \cdot d x = Φ (γ (1)) - Φ (γ (0))

. In mechanics a conservative force is conventionally written

- \nabla V

, so its potential energy is

V = - Φ

. This spatial potential differs from the scalar potential

V (ϕ)

that denotes the derivative-free energy density of a scalar field.

Divergence theorem

Fundamental theorem for line integrals

Path independence

A line integral is path independent when its value depends only on the two endpoints. On a connected domain, this holds for the line integral of every exact differential.

Stokes theorem

Stokes flux through an annular paraboloid ( $I = 3 π (R^{4} - r_{0}^{4}) /2$ )