Multivariable calculus

f : U \to R^{m}

is continuously differentiable and the segment from

x

y

lies in

U

, then

f (y) - f (x) = \int_{0}^{1} D f (x + t (y - x)) (y - x) d t .

Partial derivative ( $\partial$ )

A partial derivative measures the local response of a function to one input while its other inputs stay fixed. For a function

f

on an open subset of

R^{2}

, its partial derivatives at

(a, b)

are the limits

f_{x} (a, b) = lim_{h \to 0} \frac{f ( a + h , b ) - f ( a , b )}{h}, f_{y} (a, b) = lim_{k \to 0} \frac{f ( a , b + k ) - f ( a , b )}{k},

when the respective limits exist. Each is an ordinary derivative of a one-variable slice:

x \mapsto f (x, b)

y \mapsto f (a, y)

. This is why the other variable is treated as a constant during partial differentiation.^[ref]

For an original example, take the polynomial function

f (x, y) = x^{2} + x y + 2 y^{2} .

Holding

y

fixed gives

f_{x} = 2 x + y

, while holding

x

fixed gives

f_{y} = x + 4 y

. At

(1, 1)

, the height is

f = 4

and the two partial derivatives are $f_{x} = 3$ and $f_{y} = 5$ . The values answer different experiments on the same mathematical model: change

x

alone, or change

y

alone.

The two exact slices through that point are

f (1 + t, 1) = 4 + 3 t + t^{2}, f (1, 1 + t) = 4 + 5 t + 2 t^{2} .

Dividing either height change by

t \neq = 0

gives

3 + t

5 + 2 t

; taking the limit

t \to 0

recovers the two slopes. At

t = 0.1

, the finite height changes are

0.31

and

0.52

, although their linear approximations are

0.3

and

0.5

. A derivative measures the limiting response, rather than an exact finite change.

Figure 1.
Two coordinate slices through one point give different partial derivatives
. Original contour plot of $f = x^{2} + x y + 2 y^{2}$ , with the point $(1, 1)$ and coordinate paths. The right panel compares the exact slice height changes $3 t + t^{2}$ and $5 t + 2 t^{2}$ with their tangent line predictions $3 t$ and $5 t$ . Both slices pass through height $4$ . **Details**.

For a function of

n

variables,

\partial_{j} f (a)

varies only its

j

th coordinate: it is the same difference quotient along

e_{j}

, the

j

th standard basis vector. This gives a practical sensitivity analysis: one can compare how a mathematical model responds to changes in its individual inputs, provided the chosen system of units and the quantities held fixed are stated. Comparing the numerical sizes of two partial derivatives in different units does not by itself identify a physically preferable input.

Existence of the partial derivatives at one point is weaker than differentiability. The example under partial derivatives do not imply continuity shows how both coordinate tests can miss a bad diagonal approach. When the first partial derivatives are continuous in a neighborhood, they do give differentiability there, allowing the total differential to combine all small input changes.

Partial derivatives do not imply continuity

Define

f (x, y) = x y / (x^{2} + y^{2})

away from the origin and zero at the origin. Both partial derivatives exist everywhere and are zero at the origin, yet along

x = y

the function is

1/2

, so it is not continuous there. Coordinate-axis derivative tests do not control all neighboring directions.

Mixed partial derivative ( $f_{x y}$ )

A mixed partial derivative takes partial derivatives in different variables, for example

f_{x y} = \partial_{y} (\partial_{x} f)

. If both second partial derivatives are continuous in a neighbourhood, their order can be interchanged:

f_{x y} = f_{y x}

. The chain rule transforms mixed partial derivatives under a linear change of variables by composing the corresponding first-order differential operators.

Unequal mixed partial derivatives ( $\partial_{y} \partial_{x} f (0) = - 1, \partial_{x} \partial_{y} f (0) = 1$ )

For

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

away from the origin and zero at the origin,

∣ f ∣ \leq (x^{2} + y^{2}) /2

proves Fréchet differentiability with zero derivative. Yet

f_{x} (0, y) = - y

and

f_{y} (x, 0) = x

, giving the displayed unequal mixed partial derivatives. They exist but are not continuous at the origin, so Clairaut's theorem does not apply.

Symmetry of second derivatives ( $\partial_{i} \partial_{j} f = \partial_{j} \partial_{i} f$ )

When the second partial derivatives are continuous in a neighborhood, the order of the mixed partial derivatives can be interchanged. Expressing a rectangular finite difference as either iterated integral by the fundamental theorem of calculus and shrinking the rectangle proves the equality. Mere existence of both derivatives does not imply this symmetry.

Time derivative ( $\partial_{t}$ )

A time derivative is the partial derivative of an evolving field with respect to time while holding its spatial coordinates fixed. For a quantity depending only on time, it is an ordinary derivative.

Spatial derivative ( $\partial_{i}$ )

A spatial derivative is a partial derivative with respect to a spatial coordinate.

Directional derivative ( $D_{v} f$ )

 2  0

The directional derivative tests a function along a chosen straight line, including lines that follow neither coordinate axis. For a fixed vector

v

and a point

a

in an open subset of

R^{n}

, define

D_{v} f (a) = lim_{t \to 0} \frac{f ( a + t v ) - f ( a )}{t} = \frac{d}{d t} f (a + t v)_{t = 0},

if the limit exists. With a unit vector

v

, the parameter

t

measures signed distance in the input Euclidean space, so the directional derivative is change per unit distance. For a general

v

, it is change per unit parameter; the norm of

v

also sets the rate of travel.^[ref]

For a differentiable real-valued function, the multivariable chain rule gives

D_{v} f (a) = \nabla f (a) \cdot v .

Thus one gradient predicts the local response in every direction. This formula requires differentiability, rather than merely separate partial derivatives.^[ref]

Return to

f (x, y) = x^{2} + x y + 2 y^{2}

a = (1, 1)

. Its gradient is

(3, 5)

. In the unit direction

u = (3/5, 4/5)

D_{u} f (a) = 3 \frac{3}{5} + 5 \frac{4}{5} = \frac{29}{5} .

One can check this without invoking the gradient formula: substitution gives

f (1 + \frac{3}{5} t, 1 + \frac{4}{5} t) = 4 + \frac{29}{5} t + \frac{53}{25} t^{2} .

(4)

The difference quotient is

29/5 + (53/25) t

, whose limit is

29/5

. This calculation separates the exact change along a finite path from its first-order rate at the starting point.

Changing the travel speed changes the parameter rate. Taking

v = (3, 4) = 5 u

gives

D_{v} f (a) = 29

, because the input travels five distance units per parameter unit. For a differentiable function,

D_{c v} f (a) = c D_{v} f (a)

for any real

c

: negative

c

reverses the orientation as well as scaling the rate. Normalizing

v

before comparing different slopes avoids confusing direction with speed.

Figure 1.
Directional derivatives compare directions; a total differential predicts small finite changes
. Left: for $f = x^{2} + x y + 2 y^{2}$ at $(1, 1)$ , the rate in the unit direction $(cos θ, sin θ)$ is $3 cos θ + 5 sin θ$ . Right: along the displacement $s (0.02, - 0.01)$ , the exact height change is $0.01 s + 0.0004 s^{2}$ , compared with its linear approximation $0.01 s$ . All curves are exact calculations for the example, rather than measured data. **Details**.

The Cauchy-Schwarz inequality bounds the rate in any unit direction by

∥\nabla f (a) ∥

. Here the largest rate is

34 \approx 5.831

, attained in the direction

(3, 5) / 34

; the chosen

(3/5, 4/5)

gives the slightly smaller rate

5.8

. The perpendicular unit direction

(- 5, 3) / 34

gives zero. That means no first-order height change, rather than no change at all: along this direction the example has

f (a + t u) = 4 + (28/34) t^{2}

This is useful for following a height or temperature scalar field, choosing a local search direction in mathematical optimization, and distinguishing a coordinate sensitivity from a response along an actual path. For a curved path, derivative along a curve uses its instantaneous tangent vector. Even existence of all directional derivatives does not guarantee one common linear approximation; directional derivatives do not imply differentiability gives a concrete counterexample.

Gâteaux derivative ( $D I (v) [h] = lim_{t \to 0} [I (v + t h) - I (v)] / t$ )

A functional on a Banach space is Gâteaux differentiable at

v

when the directional limits exist and form a bounded linear functional of

h

. Fréchet differentiability is stronger, requiring a remainder uniformly small compared with the norm of the increment. A quadratic norm bound on the Taylor remainder gives the stronger property directly. Mere existence of unrelated directional limits would not establish linear differentiability.

Finite increments of a logarithmic radial function ( $s = r_{0} (e^{Δ/2} - 1)$ )

For

T (r) = ln ∣ r ∣^{2}

, starting at radius

r_{0} > 0

and moving radially outward through distance

s

changes the function by

Δ = 2 ln (1 + s / r_{0})

. Solving yields the displayed exact distance. Its small-increment limit is

s ≃ r_{0} Δ/2

, which is the initial directional derivative estimate. That linear estimate is not an exact equality for a specified finite change.

Radial derivative ( $\partial_{r} u (x) = \nabla u (x) \cdot x /∣ x ∣$ )

Away from the origin, the radial derivative is the directional derivative along the outward radial unit vector. For a Sobolev space function it is the same scalar product with the weak gradient, defined almost everywhere. On an annulus, it has no singular direction at the origin.

Directional derivatives do not imply differentiability

Define

f (x, y) = x^{3} / (x^{2} + y^{2})

away from the origin and zero at the origin. Every directional derivative exists. Along a unit vector

n

, the derivative at the origin is

n_{1}^{3}

, which is not linear in

n

. Thus these directional derivatives cannot be the restrictions of one Fréchet derivative, and the function is not Fréchet differentiable at that point.

Directional derivative of a convex function is sublinear ( $p_{x} (y) = lim_{t ↓ 0} [f (x + t y) - f (x)] / t$ )

For a locally Lipschitz function that is convex near

x

, increasing secant slopes have a finite right limit. Positive rescaling of

t

gives positive homogeneity in

y

. The convexity bound

f (x + t (y + z)) \leq [f (x + 2 t y) + f (x + 2 t z)] /2

gives subadditivity after subtracting

f (x)

, dividing by

t

and taking the limit. Thus

p_{x}

is a finite sublinear functional, although it need not be a linear functional.

Gradient ( $\nabla f$ )

In Cartesian coordinates on Euclidean space, the gradient of a scalar-valued differentiable function is the vector of its first partial derivatives. It converts a scalar field, such as temperature or a height, into a vector field describing its local change.

For a unit vector

u

, the directional derivative is

D_{u} f = u \cdot \nabla f

. The Cauchy-Schwarz inequality gives

∣ D_{u} f ∣ \leq ∣\nabla f ∣

. At a point where

\nabla f \neq = 0

, the greatest increase per unit distance occurs along

u = \nabla f /∣\nabla f ∣

, while the opposite direction gives the greatest decrease. These are first-order statements at the current point; they do not predict how the function behaves far along a straight path.

For

f (x, y) = x^{2} + 2 y^{2}

(1, 1)

\nabla f = (2, 4)

. Moving in the

x

direction gives rate

2

and moving in the

y

direction gives rate

4

, but the steepest direction is

(1, 2) / 5

, with

D_{u} f = 25 .

The direction

(- 2, 1) / 5

has rate zero and is tangent to the level curve through that point. More generally, differentiating

f (γ (t)) = constant

along a regular level curve gives

\nabla f \cdot γ^{'} = 0

, so the gradient is its normal vector.

Figure 1.
A gradient is normal to level curves and determines the directional derivative for every unit direction
. **Details**.

Functional gradient ( $\nabla_{H} F (g)$ )

For a differentiable real-valued functional

F

on a real Hilbert space

H

, its functional gradient satisfies

D F (g) [h] = ⟨ \nabla_{H} F (g), h ⟩_{H}

for every

h \in H

. A bounded linear derivative has a unique such representative by the Riesz representation theorem. The inner product matters: a formal variational derivative need not equal the functional gradient in another Hilbert space. For

F (g) = \frac{1}{2} \sum_{i} (y_{i} - g (x_{i}))^{2}

in a Reproducing kernel Hilbert space, the reproducing property gives

\nabla_{H} F (g) = \sum_{i} (g (x_{i}) - y_{i}) k (x_{i}, \cdot)

. Substitution in the defining inner product proves this formula directly.

Gradient of a radial function ( $\nabla g (r) = g^{'} (r) r / r$ )

For

r = x_{1}^{2} + \dots + x_{d}^{2} > 0

, the multivariable chain rule gives

\nabla g (r) = g^{'} (r) \frac{( x _{1} , \dots , x _{d} )}{r} .

Indeed,

\partial r / \partial x_{j} = x_{j} / r

. Thus the gradient points radially outward when

g^{'} (r) > 0

, inward when it is negative, and vanishes when it is zero.

At the origin the displayed quotient does not apply. If

g

has a finite right derivative at zero, the radial function is differentiable there exactly when

g^{'} (0^{+}) = 0

, in which case its gradient is zero. Necessity follows by comparing opposite directional difference quotients: their linear parts cannot both equal a common nonzero radial slope. Sufficiency follows from

g (r) - g (0) = o (r)

, the definition of differentiability with zero linear part.

Gradient of a dot product ( $\nabla (F \cdot G) = (F \cdot \nabla) G + (G \cdot \nabla) F + F \times curl G + G \times curl F$ )

For differentiable vector fields, expand the cross products using the contraction of two Levi-Civita symbols. The directional-derivative terms cancel the corresponding terms in the two cross products, leaving

\partial_{i} (F_{j} G_{j})

. This proves the displayed gradient identity by the product rule.

Gradient field

A gradient field is a vector field of the form

\nabla f

for a scalar potential

f

. Its curl vanishes wherever the required second derivatives commute.

Total differential ( $df$ )

The total differential is the linear map that combines the first-order responses to simultaneous changes in all coordinates. For a differentiable real-valued function at

a

, it can be written

d f_{a} (h) = \sum_{j = 1}^{n} \partial_{j} f (a) h_{j} = \nabla f (a) \cdot h, df = \sum_{j = 1}^{n} \frac{\partial f}{\partial x _{j}} d x_{j} .

Here

h

is an input vector;

d x_{j}

denotes the coordinate linear functional

h \mapsto h_{j}

. The coordinate formula and the Fréchet derivative describe the same linear map. The defining approximation is

f (a + h) - f (a) = d f_{a} (h) + o (∥ h ∥) (h \to 0) .

The little o notation says the remaining approximation error becomes small relative to the input norm, across every direction of approach.^[ref]

For

f = x^{2} + x y + 2 y^{2}

a = (1, 1)

, $d f_{a} (h, k) = 3 h + 5 k$ . A simultaneous change

(h, k) = (0.02, - 0.01)

gives the linear approximation

Δ f \approx 3 (0.02) + 5 (- 0.01) = 0.01 .

The exact algebra is

f (1 + h, 1 + k) - 4 = 3 h + 5 k + h^{2} + hk + 2 k^{2} .

(4)

Consequently the exact change is

0.0104

, and the approximation error is

0.0004

. If both input changes are halved, the linear prediction becomes

0.005

and the error becomes

0.0001

: the first-order response halves, while the quadratic error falls by a factor of four. The right panel under directional derivative displays this comparison as the displacement is scaled.

For a quick check of differentiability,

∣ hk ∣ \leq (h^{2} + k^{2}) /2

gives

∣ h^{2} + hk + 2 k^{2} ∣ \leq \frac{5}{2} (h^{2} + k^{2}) .

(5)

Dividing by

h^{2} + k^{2}

makes this bound tend to zero. The same calculation proves that the tangent plane to the graph of a function at

(1, 1, 4)

z = 4 + 3 (x - 1) + 5 (y - 1)

. It approximates the entire local surface, rather than only two coordinate slices.

The total differential is useful for estimating the combined effect of small input changes without reevaluating a complicated function. For example, if

∣ h ∣ \leq 0.02

and

∣ k ∣ \leq 0.01

, then

∣ d f_{a} (h, k) ∣ \leq 0.11

. The exact remainder bound from the displayed polynomial is at most

0.0008

, so this example has

∣Δ f ∣ \leq 0.1108

throughout that input rectangle. These are deterministic bounds; a variance calculation additionally needs a joint probability distribution and its covariances. A first-order estimate is also local: large changes require control of the remainder, not merely knowledge of the partial derivatives at one point.

Hessian matrix ( $Hess f$ )

The Hessian matrix describes how the slope of a function changes in different directions. For a real-valued function

f

with continuous second partial derivatives on an open subset of

R^{n}

, it is

H_{f} (x) = (\frac{\partial ^{2} f}{\partial x _{i} \partial x _{j}} (x))_{i, j = 1}^{n} .

The symmetry of second derivatives makes

H_{f}

a symmetric matrix under these hypotheses. It is also the Jacobian matrix of the gradient: near

x

, a small displacement

h

changes

\nabla f

H_{f} (x) h + o (∥ h ∥)

. Thus the gradient measures the current slope, while the Hessian measures its local variation. ^[ref]

For example, take the quadratic form

q (x, y) = x^{2} + 2 x y + 3 y^{2} .

Its gradient and Hessian matrix are

\nabla q = (2 x + 2 y, 2 x + 6 y)^{T}, H_{q} = (2226) .

The only critical point is the origin. Completing the square gives

q = (x + y)^{2} + 2 y^{2}

, so the origin is a strict global minimum. The eigenvalues

4 - 22

and

4 + 22

are both positive, recovering the same conclusion from the positive definiteness of the Hessian. Its eigenvectors identify the rotated directions in which the quadratic form has its smallest and largest bending.

Figure 1.
Positive, negative and mixed Hessian eigenvalues produce different critical-point geometry
. Contour plots of $q = x^{2} + 2 x y + 3 y^{2}$ , $- q$ , and $s = x^{2} - y^{2}$ . All three have a critical point at the origin; the first two have elliptical contours, while the last has positive and negative sectors meeting along its zero contours. **Details**.

For a fixed vector

v

, restriction to a straight line gives the directional interpretation

\frac{d ^{2}}{d t ^{2}} f (x + t v)_{t = 0} = v^{T} H_{f} (x) v .

(4)

This is the quadratic form associated with the Hessian matrix. For a unit eigenvector

v

, it equals the corresponding eigenvalue. Opposite signs therefore mean that some directions bend upwards and others downwards. The second-derivative test turns this information into a classification when the gradient vanishes.

The same object is useful for computation. Newton's method in optimization chooses a displacement

h

by solving

H_{f} (x) h = - \nabla f (x)

when the Hessian matrix is an invertible matrix; this makes the gradient of the local quadratic model vanish. For the displayed

q

, the equation gives

h = - (x, y)^{T}

, so one exact step reaches the minimum from any starting point. For a general function, a Hessian with opposite-sign eigenvalues or a large condition number need not give a useful minimizing step, and the local minimum need not be a global minimum.

The name honors Otto Hesse. ^[ref] An older geometric use is the Hessian criterion for flexes of a plane cubic, connecting a determinant of second partial derivatives with inflection points of an algebraic curve. Hesse's 1844 paper Über die Wendepuncte der Curven dritter Ordnung is an early source on that geometric problem. ^[ref]

Hessian at a critical point under a change of coordinates ( $H_{f \circ ϕ} (u_{}) = J_{ϕ} (u_{})^{T} H_{f} (ϕ (u_{})) J_{ϕ} (u_{})$ )

Let

ϕ

be a twice continuously differentiable local diffeomorphism, and suppose

\nabla f (ϕ (u_{*})) = 0

. The chain rule gives the displayed transformation of the Hessian matrix: terms containing second partial derivatives of

ϕ

are multiplied by first partial derivatives of

f

and vanish at the critical point. By Sylvester's law of inertia, the numbers of positive, negative and zero eigenvalues are unchanged. Independently of nondegeneracy, the local diffeomorphism preserves the critical point's status as a local minimum, local maximum or saddle point of a scalar function, because it maps neighbourhoods to neighbourhoods.

Matrix-free Hessian-vector product ( $v \mapsto \nabla^{2} l (x) v$ )

Applying a Hessian matrix to a vector without explicitly storing all its entries. For

H = A^{T} Δ A

with diagonal

Δ

, evaluate

H v = A^{T} (Δ (A v))

. This uses two matrix-vector multiplications and a diagonal scaling:

O (n d)

arithmetic for dense

A \in R^{n \times d}

, rather than forming the

d \times d

Hessian matrix. Iterative second-order methods can use this operation, with safeguards for indefinite curvature when the objective is nonconvex.

Hessian of a sum of functions of linear forms ( $\nabla^{2} l (x) = A^{T} diag (y_{i} f^{''} (a_{i}^{T} x)) A$ )

For twice differentiable

f : R \to R

, constant vectors

a_{i}

and constant scalars

y_{i}

, let

l (x) = \sum_{i} y_{i} f (a_{i}^{T} x)

, and let row

i

A

a_{i}^{T}

. The chain rule gives

\nabla l (x) = \sum_{i} y_{i} f^{'} (a_{i}^{T} x) a_{i}, \nabla^{2} l (x) = \sum_{i} y_{i} f^{''} (a_{i}^{T} x) a_{i} a_{i}^{T} .

The second formula is the displayed matrix product because entry

(j, k)

is the same weighted sum

\sum_{i} y_{i} f^{''} (a_{i}^{T} x) a_{ij} a_{ik}

. Signed diagonal factors do not ensure a positive-definite Hessian matrix. The factorization also gives an efficient matrix-free Hessian-vector product.

Hessian determinant ( $det D^{2} f$ )

The determinant of the Hessian matrix. For a twice continuously differentiable function of two variables, it is

f_{xx} f_{yy} - f_{x y}^{2}

. A negative value at a critical point implies a saddle point; a positive value combines with the sign of

f_{xx}

in the second-derivative test. A zero value is inconclusive.

Quadratic growth from a bounded Hessian ( $∥ D^{2} F ∥_{\infty} < \infty ⟹ ∣ F (p) ∣ \leq C (1 + ∣ p ∣^{2})$ )

For

F \in C^{2} (R^{n})

with bounded Hessian matrix, Taylor's formula along the line from zero to

p

gives

∣ F (p) ∣ \leq ∣ F (0) ∣ + ∣ D F (0) ∣∣ p ∣ + \frac{1}{2} ∥ D^{2} F ∥_{\infty} ∣ p ∣^{2}

. Its gradient obeys

∣ D F (p) ∣ \leq ∣ D F (0) ∣ + ∥ D^{2} F ∥_{\infty} ∣ p ∣

. Consequently

\int_{Ω} F (D u)

is finite for

u \in H^{1} (Ω)

when

Ω

has finite measure, and

D F (D u) \in L^{2}

Higher-order test for separated extrema

For

f (x, y) = g (x) + h (y)

, a degenerate Hessian matrix can be supplemented by the first nonzero term of each one-variable displacement. Two negative leading even powers give a strict local maximum, two positive ones a strict local minimum, and opposite signs give a saddle. Looking along the coordinate directions and bounding the separated terms avoids an inconclusive quadratic test.

For an explicit contrast, consider

f_{+} (x, y) = x^{4} + y^{4}, f_{-} (x, y) = x^{4} - y^{4} .

Both have zero gradient and the same zero Hessian matrix at the origin:

H_{f_{+}} (0, 0) = H_{f_{-}} (0, 0) = (0000) .

Nevertheless,

f_{+}

is positive at every nonzero point, so the origin is a strict global minimum. For

f_{-}

, the restrictions

f_{-} (t, 0) = t^{4} > 0

and

f_{-} (0, t) = - t^{4} < 0

for

t \neq = 0

show that the origin is a saddle point of a scalar function. Identical quadratic information can conceal different geometry when higher-order terms are decisive.

Figure 1.
The same zero Hessian can occur at a strict minimum or a saddle
. The quartic functions $x^{4} + y^{4}$ and $x^{4} - y^{4}$ both have zero second partial derivatives at the origin. Their contour geometry and signs differ, so a singular Hessian matrix alone cannot classify the critical point. **Details**.

Negative semidefinite matrix

A symmetric matrix

A

is negative semidefinite when

v^{T} A v \leq 0

for every vector

v

. Equivalently, all its eigenvalues are nonpositive.

Second-derivative test

Let

f

have continuous second partial derivatives near an interior critical point

x_{*}

of an open subset of

R^{n}

, with

\nabla f (x_{*}) = 0

. The Hessian matrix gives the following sufficient conditions:

All eigenvalues positive: a strict local minimum.
All eigenvalues negative: a strict local maximum.
Both positive and negative eigenvalues: a saddle point of a scalar function.

A positive semidefinite or negative semidefinite Hessian matrix with a zero eigenvalue does not settle the classification. The higher-order test for separated extrema gives two quartic examples with exactly the same zero Hessian. These conditions apply to an interior critical point; a boundary optimum or a constrained optimization problem requires attention to the allowed directions. ^[ref]

The proof outline is the Taylor expansion

f (x_{*} + h) - f (x_{*}) = \frac{1}{2} h^{T} H_{f} (x_{*}) h + o (∥ h ∥^{2}) .

For a positive-definite matrix, the smallest eigenvalue supplies a positive bound proportional to

∥ h ∥^{2}

, which dominates the remainder for small enough

h

. Negate the function for the local maximum case. For opposite-sign eigenvalues, choose the two corresponding eigenvectors: the leading terms have opposite signs along their lines.

There are also necessary conditions. At an interior local minimum, the Hessian matrix is positive semidefinite; at an interior local maximum, it is negative semidefinite. Restriction to each straight line reduces this to the one-variable second derivative condition. Since the Laplacian is the trace of the Hessian, it is nonnegative at such a local minimum and nonpositive at such a local maximum. The sign of the Laplacian alone is insufficient:

x^{2} - 2 y^{2}

has negative Laplacian but a saddle point of a scalar function at the origin.

Existence of second partial derivatives does not justify the Hessian test

Define

f (0, 0) = 0

and, elsewhere,

f (x, y) = x^{2} + y^{2} - \frac{6 x ^{2} y ^{2}}{x ^{2} + y ^{2}} .

All first and second partial derivatives exist at the origin:

f_{x} = f_{y} = f_{x y} = f_{y x} = 0

and

f_{xx} = f_{yy} = 2

. Thus its Hessian matrix there is the positive-definite matrix

2 I

. Nevertheless

f (t, 0) = t^{2} > 0

and

f (t, t) = - t^{2} < 0

for

t \neq = 0

, so the origin is a saddle point of a scalar function. A valid second-derivative test needs a second-order Taylor expansion with a remainder

o (x^{2} + y^{2})

; continuous second partial derivatives in a neighborhood are a sufficient hypothesis. Existence of those derivatives at the point alone does not supply this hypothesis.

Critical points escaping to infinity in a Gaussian-weighted bilinear function ( $f_{α} (x, y) = x y e^{- α (x^{2} + y^{2})}$ )

The origin is a saddle point of a scalar function for every real parameter. For

α > 0

four further critical points have

x^{2} = y^{2} = 1/ (2 α)

. Equal-sign coordinates give global maxima and opposite-sign coordinates give global minima, with values

\pm 1/ (2 e α)

. The Hessian matrix there is

- 4 αx y e^{- 1} I

. Their distance from the origin is

α^{- 1/2}

, so as

α ↓ 0

they escape every compact region. At nonpositive

α

the origin is the only critical point. This illustrates why the number of critical points on an unbounded domain can change without a finite degenerate critical point appearing.

Nondegenerate critical point

A critical point of a real function with continuous second partial derivatives is nondegenerate when its Hessian matrix is an invertible matrix there, equivalently when none of its eigenvalues is zero. A positive-definite matrix gives a strict local minimum, a negative definite Hessian gives a strict local maximum, and opposite-sign eigenvalues give a saddle point of a scalar function. A nondegenerate critical point need not be an extremum:

x^{2} - y^{2}

has Hessian matrix

diag (2, - 2)

, which is an invertible matrix.

Such critical points are the local building blocks of Morse theory.

Vector calculus

Vector calculus studies differentiation and integration of scalar fields and vector fields. Three operators answer different local questions: the gradient describes change of a scalar value, the divergence describes net outward flow, and the curl describes local circulation. Their associated integral theorems turn local derivatives into measurements along paths and across boundaries.

The difference between divergence and curl is especially visible in two smooth vector fields on three-dimensional Euclidean space:

F (x, y, z) = (x, y, 0), G (x, y, z) = (- y, x, 0) .

At each point they have the same Euclidean norm

x^{2} + y^{2}

, but their directions differ. For

F

\nabla \cdot F = 2

and

\nabla \times F = 0

. For

G

\nabla \cdot G = 0

and

\nabla \times G = (0, 0, 2)

. One expands radially in each horizontal plane; the other describes rigid rotation.

Figure 1.
Equal-magnitude fields can have different divergence and curl: radial expansion gives outward flux, while rotation gives circulation
. **Details**.

Around the counterclockwise unit circle in the plane

z = 0

F

is normal to the boundary and has zero tangential component. Its outward planar flux integral is

2 π

, while its line integral around the circle is zero. For

G

the values are reversed: outward planar flux is zero, and circulation is

2 π

. The divergence and curl predict exactly which boundary measurement survives.

Why is this useful? A velocity field can move material through a boundary or circulate it within a region. Conservation of mass connects its divergence with density changes through the continuity equation; its curl is vorticity. In electromagnetism, the Maxwell equations use the same operators to connect electric fields, magnetic fields, and their sources. The common mathematical language lets the same integral calculations work across different physical models. ^[ref].

The geometric framework behind these operators is developed in differential forms. The Generalized Stokes theorem unifies the fundamental theorem for line integrals, Green theorem, Stokes theorem, and divergence theorem. In coordinates their formulas look different, but each relates accumulation over a region to a measurement on its boundary.

Wilson's Vector Analysis, published in 1901, was based on J. Willard Gibbs's lectures at Yale. Wilson's preface explains that examples from geometry, mechanics, and physics are interspersed with the mathematical development, so readers encounter applications as soon as they have the tools to understand them. That is a useful way to approach vector calculus: learn each operation through a measurement it makes possible. ^[ref].

Vector calculus identity

Divergence of a curl ( $\nabla \cdot (\nabla \times A) = 0$ )

For a twice continuously differentiable vector field in Cartesian coordinates, expand the curl and then its divergence. Every mixed partial derivative occurs twice with opposite signs, proving the identity. In electromagnetism this forces the right side of the Ampère-Maxwell equation to be divergence-free, yielding local charge conservation.

Euclidean scalar-gradient stress identity ( $\partial_{k} Σ_{kj} = (Δ ψ) \partial_{j} ψ$ )

For a sufficiently smooth real scalar field on Euclidean space, set

Σ_{kj} = \partial_{k} ψ \partial_{j} ψ - δ_{kj} ∣\nabla ψ ∣^{2} /2

. The product rule and equality of mixed partial derivatives give

\partial_{k} Σ_{kj} = (Δ ψ) \partial_{j} ψ + \partial_{k} ψ \partial_{k} \partial_{j} ψ - \partial_{k} ψ \partial_{j} \partial_{k} ψ = (Δ ψ) \partial_{j} ψ .

The divergence theorem, applied twice, consequently gives the useful pure-boundary identity

\int_{V} r \nabla \cdot [(Δ ψ) \nabla ψ] d V = \int_{\partial V} [r (Δ ψ) \partial_{n} ψ - (\partial_{n} ψ) \nabla ψ + \frac{1}{2} ∣\nabla ψ ∣^{2} \hat{n}] d S .

Here

\partial_{n} ψ = \hat{n} \cdot \nabla ψ

uses the outward unit normal. This is the Euclidean zero-potential counterpart of the scalar stress-energy divergence identity.

Vector Analysis (book)

Dot-product gradient identity ( $\nabla (F \cdot G) = (F \cdot \nabla) G + (G \cdot \nabla) F + F \times (\nabla \times G) + G \times (\nabla \times F)$ )

For continuously differentiable vector fields, the gradient of their dot product expands as displayed. The contraction of two Levi-Civita symbols gives

[F \times (\nabla \times G)]_{i} = F_{j} \partial_{i} G_{j} - F_{j} \partial_{j} G_{i}

; adding the directional derivative cancels the second term. Interchanging the fields and using the product rule proves the identity. It does not require either field to be irrotational.

Helmholtz decomposition ( $v = \nabla ϕ + w$ )

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.

Divergence of a cross product

For differentiable vector fields

A

and

B

\nabla \cdot (A \times B) = B \cdot (\nabla \times A) - A \cdot (\nabla \times B) .

Vector field

 2  0

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

Radial vector field ( $F (r) = f (r) r$ )

A vector field directed along the radius from a fixed origin, with magnitude depending only on radius. In three-dimensional Euclidean space and away from the origin,

F = f (r) r = r f (r) \hat{r}

. The product rule gives

\nabla \cdot F = 3 f + r f^{'}

, and the curl vanishes because the derivative matrix is symmetric. These identities alone do not establish regularity at the origin.

Continuously differentiable vector field ( $F \in C^{1} (Ω; R^{n})$ )

A vector field on an open subset of Euclidean space whose component functions have continuous first partial derivatives. Its Jacobian matrix is continuous; this is a standard regularity assumption for the divergence theorem and Stokes theorem.

Flow of a vector field ( $\frac{d}{d t} Φ_{t} (p) = V (Φ_{t} (p))$ )

The local flow of a smooth vector field is the family of solutions of its ordinary differential equation, with

Φ_{0} (p) = p

. Uniqueness gives

Φ_{s + t} = Φ_{s} \circ Φ_{t}

wherever defined. A compactly supported smooth vector field on a manifold without boundary is complete, so its flow exists for all real times.

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.

Radial power-law flux through a spherical shell ( $\int_{\partial V} F \cdot d A = 4 π (R^{2 - γ} - ϵ^{2 - γ})$ )

Let

V = {ϵ < r < R}

with

ϵ > 0

and

F = r / r^{γ}

. Direct differentiation gives

\nabla \cdot F = (2 - γ) r^{- γ - 1}

. For

γ \neq = 2

, integration against

4 π r^{2} d r

gives the displayed expression. Independently, the two boundary spheres contribute

4 π R^{2 - γ}

and

- 4 π ϵ^{2 - γ}

, since the outward normal of the shell at its inner boundary points toward the origin. For

γ = 2

, the divergence is zero in the shell and the surface contributions cancel exactly. The excluded origin must not be silently added to this regular-domain calculation.

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

Antisymmetric first moment of a localized solenoidal field ( $\int x_{i} J_{j} d V = - \int x_{j} J_{i} d V$ )

For a solenoidal vector field

J

with compact support and suitable integrability, apply the divergence theorem to

x_{i} x_{j} J

. It gives zero boundary term and hence the displayed antisymmetry. Equivalently, for any constant vector

C

, the vector triple product identity yields

\int (C \cdot r) J d V = \frac{1}{2} C \times \int (J \times r) d V

. A piecewise smooth field must have no normal jump at its support boundary; otherwise its weak divergence contains a surface contribution and the argument fails.

Solenoidal flux depends on the oriented boundary ( $\int_{S_{1}} F \cdot d S = \int_{S_{2}} F \cdot d S$ )

Suppose two oriented surfaces have the same oriented boundary and

S_{1} \cup (- S_{2})

bounds a volume inside a domain where the vector field

F

is smooth and solenoidal. The divergence theorem gives zero flux through their closed union, proving the displayed equality. If a global vector potential

F = \nabla \times G

exists on the relevant surfaces, the Stokes theorem proves the equality directly from their common boundary. The domain condition matters: a singularity inside the joining volume can obstruct the divergence argument.

Poloidal-toroidal decomposition ( $F = \nabla \times \nabla \times (S x) + \nabla \times (T x)$ )

A spherical poloidal-toroidal decomposition represents a solenoidal vector field with zero flux through each sphere by scalar potentials

S, T

. The second term is tangent to concentric spheres; the first carries the radial component. Requiring zero angular means for the potentials fixes their otherwise irrelevant radial additions. In a plane layer, the analogous fluctuating components are

\nabla \times \nabla \times (ϕ \hat{z})

and

\nabla \times (ψ \hat{z})

, with any horizontally uniform components handled separately. These constructions enforce the divergence-free condition because the divergence of a curl vanishes.

// Target: fluid-mechanics.bigb

Poloidal magnetic energy bound by the radial scalar gradient ( $\int ∣ B_{P} ∣^{2} \leq \frac{1}{2} \int ∣\nabla P ∣^{2}$ )

For an isolated solenoidal magnetic field, zero flux through every sphere removes the degree-zero spherical harmonic of

P = B \cdot x

. In a poloidal-toroidal decomposition, let

L^{2} = - Δ_{S^{2}}

. The spherical poloidal component obeys

\int_{R^{3}} ∣ B_{P} ∣^{2} = \int_{R^{3}} \nabla (L^{- 2} P) \cdot \nabla P .

P = \sum_{l \geq 1, m} p_{l m} (r) Y_{l m}

, put

D_{l m} = \int_{0}^{\infty} [r^{2} ∣ p_{l m}^{'} ∣^{2} + l (l + 1) ∣ p_{l m} ∣^{2}] d r

. The two sides of the bound are respectively

\sum D_{l m} / [l (l + 1)]

and

\frac{1}{2} \sum D_{l m}

. Since

l (l + 1) \geq 2

, the bound follows, with equality for degree-one radial scalars. Together with the radial-flow dynamo energy bound it gives

(q / η)^{2} \geq 2 E_{P} / E_{total}

for a marginal or growing dynamo field.

// Target: fluid-mechanics.bigb

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} .

For a continuously differentiable vector field, the divergence is the limiting outward flux per unit volume through a small closed surface. Opposite faces of a small rectangular box almost cancel; the remaining differences in the three components give the sum of partial derivatives above. The divergence theorem extends this local balance to a whole region.

For

F = (x, y, 0)

, the divergence is

1 + 1 + 0 = 2

. In a planar unit square, the right edge contributes outward flux

1

and the top contributes

1

, while the left and bottom contribute zero. The total

2

equals the integral of the planar divergence over area

1

F

is a velocity field, positive divergence means local volume expansion, not automatically creation of mass. The continuity equation gives

D ρ / D t = - ρ \nabla \cdot F

, where

D / D t

is the material derivative. For this field, a uniform initial density evolves as

ρ (t) = ρ (0) e^{- 2 t}

while material volumes grow as

e^{2 t}

. Their product stays constant.

Horizontal flow convergence ( $C = - \nabla_{h} \cdot u$ )

Horizontal flow convergence is the negative horizontal divergence of a velocity field. Positive convergence reduces the horizontal area of a material patch; for constant-depth linear shallow water equations, the surface equation is

η_{t} = H C

, so convergence raises the free surface and divergence lowers it. This is a fluid-flow notion, distinct from convergence of a numerical sequence.

Weak divergence ( $⟨ \nabla \cdot J, φ ⟩ = - \int J \cdot \nabla φ d V$ )

The divergence defined through weak derivatives: pair a locally integrable vector field

J

with a smooth compactly supported test function

φ

using the displayed integration-by-parts identity. A piecewise smooth field can have a surface contribution where its normal component jumps. Thus zero classical divergence on each side does not imply zero weak divergence across the interface.

Velocity-space divergence ( $\nabla_{v} \cdot A$ )

The divergence with respect to velocity coordinates, with position held fixed. In

d

dimensions,

A (x, v) = B (x) - a (x) v

has

\nabla_{v} \cdot A = - d a

. A kinetic conservation law must retain this term for velocity-dependent forces.

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.

Inverse-square radial point source ( $\nabla \cdot (\hat{r} / r^{2}) = 4 π δ^{(3)} (r)$ )

The inverse-square radial vector field has zero ordinary divergence at every nonzero radius but flux

4 π

through every centred sphere. It is locally integrable in three dimensions. For a smooth compactly supported test function

η

, integrate by parts outside a ball of radius

ϵ

. The inward-oriented inner boundary gives

- \int F \cdot \nabla η = \int_{S^{2}} η (ϵ \hat{r}) d Ω

, which tends to

4 π η (0)

. This proves the displayed distributional divergence, with the three-dimensional Dirac delta function representing the point source.

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.

Total divergences in a variational principle

Adding

d F (q, t) / d t

to a mechanical Lagrangian changes its action by

F (q_{f}, t_{f}) - F (q_{i}, t_{i})

. Fixed endpoint variations therefore leave its bulk Euler-Lagrange equations unchanged. For fields, adding

\partial_{μ} K^{μ}

changes the action by a boundary flux. The same conclusion holds for compactly supported variations or appropriate fixed boundary data. If

K

depends on field derivatives, those boundary data must also control their variations. Such terms can still change canonical momenta, boundary observables, and representatives of Noether currents; equivalence of bulk equations does not mean that every boundary-value problem is identical.

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} .

For a continuously differentiable vector field in three-dimensional Euclidean space, the component of its curl along a unit normal vector is the limiting circulation per unit area around a small loop in the perpendicular plane. The loop direction and the normal obey the right-hand rule. Stokes theorem gives the corresponding relation for a finite surface.

For the rigidly rotating velocity field

G = (- Ω y, Ω x, 0)

, direct differentiation gives

\nabla \times G = (0, 0, 2Ω)

. A circle of radius

R

has tangential speed

Ω R

and circumference

2 π R

, so its circulation is

2 π Ω R^{2}

, twice the angular velocity times the enclosed area. The factor of two distinguishes vorticity from rigid-body angular velocity.

Curved integral curves alone do not demonstrate nonzero curl. The field

H = (x, - y, 0)

has zero curl, yet its trajectories satisfy

x = x_{0} e^{t}

y = y_{0} e^{- t}

and hence

x y = x_{0} y_{0}

: away from the axes they are hyperbolas. The curl measures local circulation of the field, not the curvature of an individual trajectory.

Volume integral of a curl ( $\int_{V} \nabla \times G d V = \int_{\partial V} n \times G d S$ )

The divergence theorem applied to

G \times c

for every constant vector

c

gives

\int_{V} \nabla \times G d V = \int_{\partial V} n \times G d S

, with outward unit normal

n

. Indeed

\nabla \cdot (G \times c) = c \cdot (\nabla \times G)

, while

(G \times c) \cdot n = c \cdot (n \times G)

. Equality of the scalar projections for every

c

proves the vector identity. Reversing the surface cross product reverses the sign.

Curl product rule for a scalar field ( $\nabla \times (ϕ F) = ϕ \nabla \times F + \nabla ϕ \times F$ )

For a continuously differentiable scalar field

ϕ

and vector field

F

, the curl satisfies

\nabla \times (ϕ F) = ϕ \nabla \times F + \nabla ϕ \times F .

Indeed, with fixed Cartesian basis vectors

e_{i}

, the Cartesian expression for the curl and the product rule give

\sum_{i} e_{i} \times \partial_{i} (ϕ F) = \sum_{i} (\partial_{i} ϕ) e_{i} \times F + ϕ \sum_{i} e_{i} \times \partial_{i} F .

The first sum is the cross product of the gradient of

ϕ

with

F

; the second is

ϕ

times the curl of

F

. The formula is useful when applying the Stokes theorem to a weighted vector field.

Curl of a gradient ( $\nabla \times \nabla f = 0$ )

For a twice continuously differentiable function, the curl of its gradient vanishes. In components,

ϵ_{ijk} \partial_{j} \partial_{k} f = 0

because the second derivatives are symmetric in

j, k

while the Levi-Civita symbol is antisymmetric. Thus adding a gradient to a vector potential preserves the represented curl. No topological assumption is needed for this identity, unlike the converse assertion that every curl-free vector field has a global potential.

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 Poincaré 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.

Velocity reconstruction from a mixed-boundary vector Poisson problem ( $Δ A = - \nabla \times u, \nabla \cdot A ∣_{\partial D} = 0, n \cdot \nabla \times A = n \cdot u ⟹ \nabla \times A = u$ )

Let

D

be bounded, connected and simply connected, and let

u

be a smooth divergence-free field. Suppose a sufficiently smooth vector field

A

satisfies the displayed Poisson and boundary conditions. Divergence gives

Δ div A = 0

with zero boundary value, so harmonic uniqueness gives

div A = 0

. The vector Laplacian identity then implies

\nabla \times (\nabla \times A - u) = 0

On a simply connected domain, define a scalar potential by integrating this curl-free difference along a path from a fixed point. Vanishing circulation around contractible loops proves path independence. Thus the difference is

\nabla χ

; its divergence and boundary normal component give

Δ χ = 0

and

\partial_{n} χ = 0

. Green's identity yields

\int_{D} ∣\nabla χ ∣^{2} = 0

, proving the reconstruction. Additional conditions on

A

can specify its own boundary problem; the argument assumes its existence and proves that its curl is exactly

u

. Simple connectivity is essential for eliminating possible non-gradient harmonic circulation fields.

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 takes the divergence of a scalar field's gradient. For a real-valued function

f

on an open region of Euclidean space, with continuous partial derivatives through order two, its expression in a Cartesian coordinate system is

Δ f = \nabla \cdot \nabla f = \sum_{i = 1}^{n} \frac{\partial ^{2} f}{\partial x _{i}^{2}} = tr (D^{2} f) .

Thus the Laplacian, also called the Laplace operator, adds the coordinate second derivatives; equivalently it is the matrix trace of the Hessian matrix. Its output is another scalar field.^[ref]

Three original examples distinguish shape from this sum. The bowl

f (x, y) = x^{2} + y^{2}

has

Δ f = 2 + 2 = 4

. The saddle

g (x, y) = x^{2} - y^{2}

instead has

Δ g = 2 - 2 = 0

: opposite second derivatives cancel. It is a harmonic function, although its graph of a function is not a plane. For the Gaussian function

h (x, y) = e^{- r^{2}}

, with

r^{2} = x^{2} + y^{2}

, differentiation gives

Δ h = 4 (r^{2} - 1) e^{- r^{2}} .

Its Laplacian is negative inside the circle of radius one, zero on that circle, and positive outside it. The Gaussian function itself is positive throughout: its value and its Laplacian describe different properties.

Figure 1.
The same Laplacian measures a bowl, a saddle and a Gaussian function
. Original analytic surfaces. Height shows $f$ ; the common color scale shows $Δ f$ , with blue negative, neutral gray zero and red positive. Vertical scales differ. The black circle on the Gaussian function is $r = 1$ , where its Laplacian changes sign. **Details**.

The sum does not favor a particular orientation. Under a change of Cartesian coordinate system represented by an orthogonal matrix

Q

, the Hessian matrix becomes

Q^{T} (D^{2} f) Q

. The cyclic property of the trace and

Q Q^{T} = I

preserve its matrix trace. This makes the Laplacian useful when the behavior depends on spatial differences equally in every direction, rather than on the arbitrary choice of axes.

For a vector field, the Euclidean Laplacian acts separately on components expressed in a fixed Cartesian coordinate system. Components in a position-dependent basis require differentiating that basis too; applying a scalar formula separately to polar components omits those terms.^[ref] The scalar formula in Laplacian in polar coordinates is valid away from

r = 0

; smoothness at the origin must be handled in coordinates that remain regular there.

Rotational invariance of the Laplacian ( $Δ_{x} (u \circ R) = (Δ u) \circ R$ )

For a constant orthogonal matrix

R

, the Laplacian is unchanged by the coordinate transformation

ξ = R x

. If

u

is twice continuously differentiable, the chain rule gives

\sum_{i} \frac{\partial ^{2} u}{\partial x _{i}^{2}} = \sum_{i, j, k} R_{ji} R_{ki} \frac{\partial ^{2} u}{\partial ξ _{j} \partial ξ _{k}} = \sum_{j} \frac{\partial ^{2} u}{\partial ξ _{j}^{2}},

because

\sum_{i} R_{ji} R_{ki} = δ_{jk}

by orthogonality of the rows. Here

δ_{jk}

is the Kronecker delta. In two dimensions, the rotation matrix

(a b - b a)

with

a^{2} + b^{2} = 1

makes the mixed partial derivatives cancel explicitly. The same argument works for reflections and translations, so the Laplacian depends on the Euclidean geometry rather than the orientation or origin of the coordinates.

Laplacian as departure from a local average ( $A_{h} f - f = \frac{1}{4} h^{2} Δ f + O (h^{4})$ )

The Laplacian detects whether a value lies below or above nearby values, to second order. For a scalar field whose partial derivatives through order four are continuous functions, take the arithmetic mean of four equally spaced neighbors:

A_{h} f (x, y) = \frac{f ( x + h , y ) + f ( x - h , y ) + f ( x , y + h ) + f ( x , y - h )}{4} .

Applying Taylor's theorem in each coordinate cancels the first- and third-order terms between opposite neighbors. The two pairs leave

A_{h} f (x, y) - f (x, y) = \frac{h ^{2}}{4} Δ f (x, y) + O (h^{4}), h \to 0.

This is an asymptotic expansion, rather than an equality for arbitrary

h

. Rearranging gives the familiar central finite difference approximation

Δ f (x, y) = \frac{f ( x + h , y ) + f ( x - h , y ) + f ( x , y + h ) + f ( x , y - h ) - 4 f ( x , y )}{h ^{2}} + O (h^{2}) .

This finite difference replaces partial derivatives by five sampled values. The arithmetic mean lies above the center for a sufficiently small

h

if the Laplacian is strictly positive there, and below if it is strictly negative. A zero Laplacian cancels this second-order discrepancy without requiring equality of individual values.

(0, 0)

, the bowl

x^{2} + y^{2}

has center value zero and four neighbor values

h^{2}

, so the discrepancy is exactly

h^{2}

. The saddle

x^{2} - y^{2}

has neighbor values

h^{2}, h^{2}, - h^{2}, - h^{2}

, whose arithmetic mean is exactly zero. For the Gaussian function

e^{- x^{2} - y^{2}}

the discrepancy is

e^{- h^{2}} - 1 = - h^{2} + O (h^{4})

, agreeing with

Δ f (0, 0) = - 4

A harmonic function need not equal this four-neighbor arithmetic mean for a finite

h

. The polynomial

f (x, y) = x^{4} - 6 x^{2} y^{2} + y^{4}

has

Δ f = 0

everywhere, but

A_{h} f (0, 0) = h^{4}

. Higher-order terms remain even when the second-order terms cancel.

At a local minimum, each coordinate second derivative is at least zero, hence the Laplacian is at least zero. At a local maximum the inequality reverses. The sum alone does not characterize an extremum: cancellation at a saddle point of a scalar function can also produce zero. The local-average interpretation explains why the heat equation smooths differences, and helps motivate the heat equation maximum principle.

Gaussian diffusion of a smooth initial profile ( $u (r, t) = (1 + 4 t)^{- 1} e^{- r^{2} / (1 + 4 t)}$ )

For diffusion with a constant diffusion coefficient

D > 0

and no sources, the model flux is

J = - D \nabla u

. A continuity equation gives

u_{t} = - \nabla \cdot J = D Δ u

. This proportionality is a constitutive assumption about transport; the continuity equation expresses conservation. If

D

varies with position, it must remain inside the divergence, giving

u_{t} = \nabla \cdot (D \nabla u)

instead.^[ref]

An exact constructed heat equation example on the whole plane uses dimensionless coordinates with

D = 1

and the smooth initial Gaussian function

u (x, y, 0) = e^{- r^{2}}

. Set

s = 1 + 4 t

. For

t \geq 0

u (x, y, t) = \frac{1}{s} e^{- r^{2} / s}, r^{2} = x^{2} + y^{2} .

Direct differentiation verifies both the initial condition and

u_{t} = Δ u = \frac{4 ( r ^{2} - s )}{s ^{3}} e^{- r^{2} / s} .

The center decreases, while the region

r^{2} > s

increases at that instant. The sign-change circle expands with radius

s

. A fixed outer location can initially increase and later decrease as this circle passes it.

Figure 1.
An exact Gaussian function spreads under the heat equation while its spatial integral stays fixed
. Original analytic model with $D = 1$ . Left: radial profiles at $t = 0, 1/4, 1$ , whose peaks are $1, 1/2, 1/5$ . Right: $u_{t} = Δ u$ at $t = 1/4$ ; the sign changes at $r = 2$ . These are model predictions, not experimental measurements. The conserved quantity uses the whole-plane area element $2 π r d r$ , rather than the unweighted area under a radial profile. **Details**.

At time

t

, the peak is

1/ s

, and the radius where the profile is

1/ e

times its peak is

s

. The width grows as the height falls. Integration in polar coordinates gives

\int_{R^{2}} u (x, y, t) d x d y = 2 π \int_{0}^{\infty} \frac{r}{s} e^{- r^{2} / s} d r = π .

Thus a decreasing peak is compatible with a conserved spatial integral. Dividing

u

π

makes a probability density function with coordinate variances

s /2 = 1/2 + 2 t

, connecting the same spreading to Brownian motion with diffusion coefficient one. Equivalently the profile is evolved by the Gaussian heat kernel with time parameter

2 t

, since that article uses generator

\frac{1}{2} Δ

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}} .

Cartesian second derivatives in polar coordinates

For

x = r cos θ

y = r sin θ

and

r > 0

, set

c = cos θ

s = sin θ

. The differential operators are

\partial_{x} = c \partial_{r} - (s / r) \partial_{θ}

and

\partial_{y} = s \partial_{r} + (c / r) \partial_{θ}

. Applying them twice gives

f_{xx} = c^{2} f_{rr} - \frac{2 sc}{r} f_{r θ} + \frac{s ^{2}}{r ^{2}} f_{θθ} + \frac{s ^{2}}{r} f_{r} + \frac{2 sc}{r ^{2}} f_{θ},

f_{yy} = s^{2} f_{rr} + \frac{2 sc}{r} f_{r θ} + \frac{c ^{2}}{r ^{2}} f_{θθ} + \frac{c ^{2}}{r} f_{r} - \frac{2 sc}{r ^{2}} f_{θ} .

Their sum is the Laplacian in polar coordinates. The first-derivative terms come from differentiating the coordinate coefficients, as required for second derivatives under a two-variable coordinate change. The origin is excluded from the coordinate formulas, even when the underlying Cartesian function is smooth there.

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 Poincaré 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}

Separated polar solutions of the biharmonic equation ( $Δ^{2} (r^{m} e^{in θ}) = (m^{2} - n^{2}) ((m - 2)^{2} - n^{2}) r^{m - 4} e^{in θ}$ )

For angular Fourier modes of integer order

n \geq 2

, the biharmonic equation in polar coordinates has radial basis

r^{n}, r^{- n}, r^{n + 2}, r^{2 - n}

. Apply the Laplacian twice to obtain the displayed indicial equation. For

n = 0

, repeated roots give the radial basis

1, lo g r, r^{2}, r^{2} lo g r

; for

n = 1

, the basis is

r^{- 1}, r, r lo g r, r^{3}

. These are Cauchy-Euler equations. Regularity at the origin and boundary conditions select the admissible powers. For

n = 2

, a constant radial factor still multiplies

cos 2 θ

sin 2 θ

, so it is bounded but fails continuity at the origin.

Complex analytic representation of biharmonic functions ( $U = Re (\overset{z}{ˉ} H (z) + G (z))$ )

H, G

are holomorphic functions, both real and imaginary parts of

\overset{z}{ˉ} H (z) + G (z)

solve the biharmonic equation. In Wirtinger derivatives,

Δ = 4 \partial_{z} \partial_{\overset{z}{ˉ}}

, so

Δ (\overset{z}{ˉ} H + G) = 4 H^{'}

is harmonic and its next Laplacian is zero. This construction gives local representations of planar solutions of the biharmonic equation; it is useful for the biharmonic stream function for planar Stokes flow.

Spherical mean of a biharmonic function ( $⟨ f ⟩_{S_{a}} = f (0) + a^{2} Δ f (0) /6$ )

For a smooth three-dimensional biharmonic function regular throughout a ball, its surface average has exactly this form. Taylor expansion and rotationally invariant spherical moments turn each even term into a multiple of an iterated Laplacian; terms from the fourth derivative onward vanish when

Δ^{2} f = 0

. This is the mean-value identity that converts the reciprocal surface average into Faxén's first law.

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 .

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

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