Geometric Calculus and the Vector Derivative

Introduction

The geometric product of a vector with a multivector splits the product into a part of lower grade and a part of higher grade, and applying that split to the differentiation of a multivector-valued function of position produces a single operator that plays the role of the gradient, the divergence, the curl and the exterior derivative at once. It is the vector derivative, written $\partial$ and defined by contracting the partial derivatives of the coordinates with the reciprocal basis. The calculus built on it is geometric calculus, and its central result is a single integral theorem from which the classical theorems of Gauss, Stokes and Cauchy, and the Cauchy integral formula of Clifford analysis, are obtained by taking grades.

This article develops the vector derivative and its fundamental theorem. The operator is not merely a notational economy: the fact that the derivative of a product obeys the Leibniz rule in the Clifford sense, and that the integral of the derivative over a region equals the integral of the function over its boundary, are the two properties from which the calculus is built, and both are proved here. The article closes with the relation to the exterior derivative and the codifferential, and with the two equations of mathematical physics that the calculus writes in a single line.

The Clifford algebra, the geometric product, the grade decomposition, the grade projection and the contraction are from Clifford Algebras and The Geometric Product and the Grade Decomposition; the reciprocal basis, the volume element and the multivectors are from Clifford Algebras in Finite Dimensions; the exterior algebra, the exterior derivative and the codifferential are from The Exterior Algebra and Differential Forms and Stokes' Theorem; the Clifford module, the monogenic functions and the Cauchy–Riemann operator are from Clifford Analysis and Clifford Modules and the Twisted Cauchy–Riemann Operator; the Lorentzian signature and the Dirac-type equation are from Dirac Differential Operators and Spinors as Minimal Left Ideals with Inner Conjugation, and the physics of the corresponding equation is in the physics menu. Nothing owned by those entries is re-derived. The space is $\mathbb{R}^n$ with the positive definite form unless a signature is named, $M$ is a compact region with smooth boundary, and $F$ is a smooth multivector-valued function on a neighbourhood of $M$.

The Vector Derivative

Definition

Definition. Let $e_1,\ldots,e_n$ be a basis of $V$ and $e^1,\ldots,e^n$ the reciprocal basis, characterised by $B(e^i,e_j)=\delta^i_j$. The vector derivative of a smooth multivector-valued function $F$ on an open set is

$$ \partial F=\sum_{i=1}^{n}e^i\,\frac{\partial F}{\partial x^i}, $$

and the derivative from the right is $F\partial=\sum_i\frac{\partial F}{\partial x^i}e^i$. The operator $\partial$ acts on the function to its right, and the factor it differentiates is marked with an overdot, as in $\dot\partial(AB)=\dot\partial A\,B+\dot A\dot\partial B$; the bracket convention is explained in the remark below.

Proposition. The vector derivative is independent of the basis used in its definition, and the two sides agree when the basis is orthonormal: $\partial=\sum_ie_i\partial_i$.

Proof. A change of basis changes both the reciprocal vectors and the coordinates in the compensating way, and the identity $B(e^i,e_j)=\delta^i_j$ is the definition of reciprocity; for an orthonormal basis the reciprocal basis is the basis itself.

Example (the derivative of the position vector). Let $x=\sum_ix^ie_i$, so that $\partial_i x=e_i$. Then

$$ \partial x=\sum_ie^i\partial_ix=\sum_ie^ie_i=n, $$

the dimension of the space; the derivative of the position vector is the dimension, which is the Clifford form of the statement that the divergence of the position field is the dimension.

The Divergence and the Curl

Theorem. For a vector-valued function $F$, the derivative splits into a scalar and a bivector part:

$$ \partial F=\partial\cdot F+\partial\wedge F, $$

where $\partial\cdot F$ is the divergence of the field, a scalar-valued function, and $\partial\wedge F$ is its curl, a bivector-valued function.

Proof. The geometric product of the vector $\partial$ with the vector $F$ has the grades $0$ and $2$, and the grade decomposition of The Geometric Product and the Grade Decomposition identifies the two parts as the contraction and the wedge; the divergence is the contraction with the derivative and the curl is the wedge.

Theorem (the Leibniz rule). For smooth functions $A$ and $B$,

$$ \dot\partial(AB)=\dot\partial A\,B+\dot A\dot\partial B, $$

where the dots indicate the factor to which the derivative is applied and the place at which the vector factor of the differentiation stands.

Proof. Differentiating the product in a coordinate and multiplying by the reciprocal vector gives $\partial_i(AB)=\partial_iA\,B+A\,\partial_iB$; multiplying by $e^i$ on the left and using the associativity of the geometric product gives the statement, the dots keeping track of which factor each differentiation acts upon.

Remark (the bracket convention). The overdot records both the factor to which the derivative is applied and the place at which the vector factor of the differentiation stands: in the second term $\dot A\dot\partial B$ the derivative acts on $B$ and the vector factor it produces stands where $A$ stands, which is what the expansion of $\partial(AB)$ in a coordinate basis produces. With this convention the Leibniz rule above is an identity and not a convention.

The Fundamental Theorem

The Statement

Theorem (the fundamental theorem of geometric calculus). Let $M$ be a compact region with smooth boundary and let $F$ be a smooth multivector-valued function on a neighbourhood of $M$. Then

$$ \int_M \dot\partial F\,dV=\int_{\partial M}dS\,F, $$

where $dV$ is the volume element of the region and $dS$ is the directed boundary element, a vector $(n)\,|dS|$ with $n$ the outward unit normal and $|dS|$ the scalar element of area.

Proof. In a coordinate system in which the region is described by the inequalities $a_i\le x^i\le b_i$ after a partition into such pieces, the left-hand side integrates the partial derivatives of $F$ over the region, and the fundamental theorem of calculus in each coordinate produces the boundary terms; the boundary terms assemble into the integral of the directed element over the boundary, because the directed element of a piece of boundary with normal $n$ is $n$ times the scalar element of area and the orientation of the region fixes the sign. The assembly is the same computation that proves the divergence theorem in vector analysis, with the Clifford product in place of the dot and cross products.

The Classical Theorems as Grades

Theorem (Gauss). For a vector-valued function $F$,

$$ \int_{\partial M}dS\cdot F=\int_M\partial\cdot F\,dV . $$

Proof. Apply the fundamental theorem to the vector-valued $F$ and take the scalar part of both sides. On the left, $dS F=dS\cdot F+dS\wedge F$ and the scalar part is $dS\cdot F$; on the right, $\dot\partial F=\partial\cdot F+\partial\wedge F$ and the scalar part is $\partial\cdot F$.

Theorem (Stokes). For a vector-valued function $F$,

$$ \int_{\partial M}dS\wedge F=\int_M\partial\wedge F\,dV . $$

Proof. Apply the fundamental theorem to the vector-valued $F$ and take the bivector part of both sides.

Theorem (the fundamental theorem of calculus). In one dimension, with $V$ spanned by the unit vector $e$ of square one and $M=[a,b]$, the fundamental theorem reads

$$ \int_a^be\,\frac{dF}{dx}\,dx=eF(b)-eF(a), $$

which is the classical statement multiplied by $e$.

Proof. The vector derivative is $e\,d/dx$, and the boundary of the interval consists of the two points with directed elements $-e$ at $a$ and $+e$ at $b$; the theorem reduces to the fundamental theorem of calculus.

Remark (the Cauchy theorem in the plane). In the plane, with $z=x+iy$ identified with the vector $x+e_1e_2y$ and with a similar identification of the derivative, the equation $\partial F=0$ is the Cauchy–Riemann equation, and the fundamental theorem becomes Cauchy's integral theorem; the full development is in Clifford Analysis, which treats the monogenic functions and the integral formula. The present article supplies the single theorem from which these are the two-dimensional cases.

The Monogenic Functions

Definition. A smooth multivector-valued function $F$ is monogenic when $\partial F=0$, and harmonic when $\partial^2F=0$.

Theorem. The vector derivative squares to the Laplacian:

$$ \partial^2=\Delta=\sum_i\frac{\partial^2}{\partial x_i^2}, $$

in the positive definite case and with orthonormal coordinates; hence every monogenic function is harmonic, in every component.

Proof. $\partial^2=\bigl(\sum_ie^i\partial_i\bigr)\bigl(\sum_je^j\partial_j\bigr)=\sum_{i,j}e^ie^j\partial_i\partial_j=\sum_i\partial_i^2$, because the mixed terms with $i\neq j$ cancel in pairs, $e^ie^j+e^je^i=2B(e^i,e^j)=0$ for the orthonormal reciprocal basis of a diagonal form, while the equal-index terms contribute $e^ie^i\partial_i^2=\partial_i^2$. A monogenic $F$ has $\Delta F=\partial(\partial F)=0$.

Remark. The converse fails: a harmonic function need not be monogenic, exactly as a harmonic function of two variables need not be holomorphic. The monogenic functions form a much smaller class, and the theory of that class, with the Cauchy integral formula, the Fischer decomposition and the function theory of the Clifford algebras, is the subject of Clifford Analysis and of Clifford Modules and the Twisted Cauchy–Riemann Operator, where the results are developed.

Remark (the Green's function). The vector derivative has a fundamental solution: up to normalisation,

$$ \partial\Bigl(-\frac{x}{S_{n-1}|x|^n}\Bigr)=\delta , $$

where $S_{n-1}$ is the area of the unit sphere in $\mathbb{R}^n$ and $\delta$ the Dirac distribution at the origin, so that the fundamental theorem applied to a region with a small sphere removed around the origin recovers the Cauchy integral formula. The verification of the distributional identity and the resulting integral formula are in Clifford Analysis.

The Relation to the Exterior Derivative

Theorem. Under the identification of a multivector-valued function with a form of the corresponding grade by the metric, the wedge part of the vector derivative is the exterior derivative and the contraction part is the codifferential:

$$ \partial\wedge F\longleftrightarrow d\omega,\qquad -\partial\cdot F\longleftrightarrow \delta\omega , $$

up to the sign convention of the codifferential.

Proof. The exterior derivative is characterised by its action on a $k$-form as the alternating part of the derivative of the components, and the wedge product of vectors with the antisymmetrisation of the derivative is exactly that; the codifferential is the formal adjoint of the exterior derivative and it is given by the contraction with the derivative, with the sign fixed by the convention of the adjoint. The identification of forms with multivectors by the metric, and the two operators, are in The Exterior Algebra and Differential Forms and Stokes' Theorem.

Corollary. In the notation of the vector derivative, the Hodge Laplacian of the exterior calculus and the square of the vector derivative agree up to the sign convention chosen for the formal adjoint:

$$ d\delta+\delta d=\pm\,\partial^2 , $$

the sign being the one carried by the definition of the codifferential, and in the flat case with the positive definite metric both are the Laplacian of the components.

Proof. Immediate from the two identifications and from the theorem on the square of the vector derivative, together with the standard sign in the definition of the codifferential recorded in Differential Forms and Stokes' Theorem.

The Equations of the Calculus

Example (the Dirac-type equation). In Lorentzian signature, with the vector derivative built from the metric of signature $(1,3)$, the equation

$$ (\partial+m)\psi=0 $$

is the Dirac equation, the object $\psi$ being a spinor of the algebra and $\partial$ the vector derivative of the space-time. The identification of the differential operator with the Clifford multiplication, the spinor module on which it acts and the physical normalisation are the subject of Dirac Differential Operators and of the physics articles of the menu.

Example (the Maxwell equation). In the same signature, the equation

$$ \partial F=J $$

with $F$ a bivector-valued field and $J$ a vector-valued current encodes the four equations of the electromagnetic field, the scalar and vector parts of the two sides being the two divergences and the two curls; the remark is that the four equations are the four grades of one equation. The details of the identification are in the physics menu, and the algebraic machinery is that of the present article and of The Geometric Product and the Grade Decomposition.

Remark (why a single operator suffices). The classical differential operators of vector analysis are three, the gradient, the divergence and the curl, and their unification in one operator is possible precisely because the geometric product of the vector $\partial$ with a multivector contains all the grades that the three operators produce. Nothing is lost in the unification: the grades of $\partial F$ can be separated, and each of the classical theorems is the corresponding grade of the single fundamental theorem, as the three theorems above show.

Summary

The vector derivative is the operator $\partial=\sum_ie^i\partial_{x^i}$, where the $e^i$ are the reciprocal basis vectors; it is basis independent and equals $\sum_ie_i\partial_i$ in an orthonormal basis. Applied to the position vector it gives the dimension of the space, $\partial x=n$, and applied to a vector-valued function it splits into the divergence and the curl, $\partial F=\partial\cdot F+\partial\wedge F$. Its Leibniz rule is $\dot\partial(AB)=\dot\partial A\,B+\dot A\dot\partial B$ with the bracket notation recording which factor is differentiated, and it squares to the Laplacian, so that the monogenic functions, those with $\partial F=0$, are harmonic; the converse fails and the theory of the monogenic functions with its integral formula is Clifford analysis.

The fundamental theorem of geometric calculus is

$$ \int_M\dot\partial F\,dV=\int_{\partial M}dS\,F, $$

the integral of the derivative of a multivector-valued function over a region equalling the integral of the function over the directed boundary. Its scalar part for a vector-valued $F$ is the divergence theorem of Gauss, $\int_{\partial M}dS\cdot F=\int_M\partial\cdot F\,dV$; its bivector part is Stokes' theorem, $\int_{\partial M}dS\wedge F=\int_M\partial\wedge F\,dV$; and in one dimension it is the fundamental theorem of calculus. In the plane, with the identification of the plane with the complex numbers, the equation $\partial F=0$ is the Cauchy–Riemann equation and the theorem is Cauchy's integral theorem. The wedge part of the vector derivative is the exterior derivative and the contraction part is the codifferential, so that $d\delta+\delta d=\Delta$; the Dirac-type equation $(\partial+m)\psi=0$ and the Maxwell-type equation $\partial F=J$ in Lorentzian signature are the two one-line forms of the equations of mathematical physics, and the four components of each are the four grades of a single equation.

Summary of Notation

Symbol	Meaning
$e^i$, $B(e^i,e_j)=\delta^i_j$	Reciprocal basis
$\partial F=\sum_ie^i\partial_iF$	Vector derivative, acting on the right
$F\partial$	Derivative from the right
$\partial x=n$	Derivative of the position vector
$\partial\cdot F$, $\partial\wedge F$	Divergence and curl of a vector field
$\dot\partial(AB)=\dot\partial A\,B+\dot A\dot\partial B$	Leibniz rule with the bracket convention
$dV$, $dS=n\,\|dS\|$	Volume element and directed boundary element
$\int_M\dot\partial F\,dV=\int_{\partial M}dS\,F$	Fundamental theorem
$\partial^2=\Delta$	Square of the vector derivative
$\partial F=0$	Monogenic functions
$d$, $\delta$	Exterior derivative and codifferential
$(\partial+m)\psi=0$, $\partial F=J$	Dirac-type and Maxwell-type equations