The exterior derivative ($d$ or $\nabla\wedge$) is a very unifying concept, in that it subsumes the gradient of a scalar, the curl of a vector, and the "divergence" of a bivector, thus also lumping together the fundamental theorem of line integrals, Stokes' Theorem, and the divergence theorem.

The interpretation of the exterior derivative might be "the extent to which the field aligns with the boundary of an infinitesimal region".

The geometric derivative seems like it should offer further unification, because, at least for vectors, it combines the interior and exterior derivatives into one. But the question is, how can I describe the meaning of the geometric derivative without resorting to describing each component separately?

On the one hand, I'm under the impression this is kind of a common issue with geometric algebra: yes, it allows you to combine objects of different degrees, yet sometimes it appears impossible to conceive of the sum as an invariant whole.

But not always. For example, you can make a "rotor" by adding a scalar and a bivector. Of course, the rotor is a transformation rather than a standalone object, but still it is a meaningful interpretation. In 3D, this is simply an element of the even sub-algebra.

Meanwhile, an even element of the 4D Clifford algebra can be construed as a transformation on bivectors, since it includes a duality rotation in addition to the spatial rotation.

So, after taking the geometric derivative of a field of degree $k$ (or perhaps mixed degree), can we articulate what it is we have? Is it some kind of transformation? Or does it represent the way that the field varies in a generalized sense? If so, what is that sense?


1 Answer 1


In Clifford Algebra to Geometric Calculus[3] section 7-2, Hestenes and Sobczyk note that the vector derivative can expressed as $$ \nabla f(x) = \lim_{\mathscr R\to 0}\frac1{|\mathscr R|I}\int_{\partial\mathscr R}\mathrm d^{n-1}x'\,f(x'). \tag{$*$} $$ (This is not the exact notation they use, but still the same thing.) This is for an $n$-dimensional space, where $I$ is the unit pseudoscalar, $\mathrm d^{n-1}x$ is the $(n-1)$-vector-valued surface measure, and $\mathscr R$ is meant to be a set containing $x$ with volume $|\mathscr R|$ and boundary $\partial \mathscr R$; the limit is meant to convey the idea that we're shrinking $\mathscr R$ around $x$.

(No, this is not rigorous; I address this somewhat further down.)

As for interpretation, I think "represent[s] the way the field varies in a generalized sense" is exactly what this says. It really is a direct generalization of the one dimensional derivative; when $n = 1$: $$ \nabla f(x) = \lim_{l\to 0}\frac1{l\cdot1}\int_{\partial[x-l/2, x+l/2]}\mathrm d^0x'\,f(x') = \lim_{l\to 0}\frac{f(x+l/2) - f(x-l/2)}l $$ Here, $\mathscr R$ is an interval $[x-l/2,x+l/2]$, $\partial\mathscr R$ is its end points, and $\mathrm d^0x'$ would be the discrete measure for the end points. We have to remember that $\mathrm d^0x'$ is an oriented measure, and the "normal vector" for $\mathscr R$ at $x-l/2$ is $-1$, and at $x+l/2$ is $1$.

A crucial difference in the general case though is the fact that $\mathrm d^{n-1}x'$ is an $(n-1)$-vector. Using the convention that $\mathrm d^{n-1}x'\,\hat N = |\mathrm d^{n-1}x'|I$ where $\hat N$ is the unit vector normal to $\partial\mathscr R$ at $x'$ (which is actually implicitly assumed by the way ($*$) is formulated), we see that $$\begin{aligned} \nabla f(x) &= \lim_{\mathscr R\to 0}\frac1{|\mathscr R|}\int_{\partial\mathscr R}|\mathrm d^{n-1}x'|\hat Nf(x') \\ &= \lim_{\mathscr R\to 0}\left[\frac1{|\mathscr R|}\int_{\partial\mathscr R}|\mathrm d^{n-1}x'|\,\hat N{\rfloor} f(x') + \frac1{|\mathscr R|}\int_{\partial\mathscr R}|\mathrm d^{n-1}x'|\,\hat N\wedge f(x')\right]. \end{aligned}$$ Recall for $A$ a blade that $\hat N{\rfloor}A\,A^{-1}$ (where ${\rfloor}$ is the left contraction[2]) is the component of $\hat N$ parallel to $A$, which we can interpret as answering the question "to what degree does $A$ contain $\hat N$?" Also recall that $\hat N\wedge A\,A^{-1}$ is the component of $\hat N$ orthogonal to $A$ in $\hat N\wedge A$, which we can interpret as answering the question "to what degree is $A$ orthogonal to $\hat N$?"

Think of the case that $\mathscr R$ is a sphere centered at $x$, and consider the subspace $\mathscr A$ parallel to $A$ that contains $x'$ (and of course $\hat N$ is normal to $\mathscr R$ at $x'$). Then $\mathscr A$ contains $x$ ("slices through $x$") iff $\hat N\wedge A = 0$ (i.e. $\hat N$ is parallel to $\mathscr A$ and $|\hat N{\rfloor}A|$ is maximal); and $\mathscr A$ is tangent at $x'$ (as $x'$ varies, $\mathscr A$ "wraps around" $x$) iff $\hat N{\rfloor} A = 0$ (i.e. $\hat N$ is orthogonal to $\mathscr A$ and $|\hat N\wedge A|$ is maximal).

So, the $\hat N{\rfloor}f(x')$ integral is telling us "how much is $f$ slicing through $x$ locally on average", and the $\hat N\wedge f(x')$ integral is telling us "how much is $f$ wrapping around $x$ locally on average." When $f$ is vector-valued, this is exactly the divergence and curl (times a pseudoscalar).

Clearly ($*$) is not rigorous; quoting Hestenes and Sobczyk:

The limit is taken by shrinking $\mathscr R$ and hence its volume $|\mathscr R|$ to zero at point $x$. We allow the limit to be proportional to a delta function or its derivatives, so the limit is well defined in the sense of distribution theory, ref. [GS]. A precise discussion of the limiting process is too involved to be given here. It requires, however, only standard arguments of analysis.

The reference [GS] is

Gelfand, I. and Shilov, G., Generalized Functions, Vol. I, Academic Press, New York (1964).

I have yet to see this definition worked out rigorously anywhere. The issue is really making the limiting process rigorous. Integration seems to be straightforward, at least in flat Euclidean space; see [1,3,5], but do take note of [4].

However, I will demonstrate an interpretation that showcases how it could recapture $\nabla = \sum_{i=1}^n e_i\frac\partial{\partial x_i}$ for an orthonormal basis $\{e_i\}_{i=0}^n$ with coordinates $\{x_i\}_{i=0}^n$.

Let $\mathcal G_n \supset \mathbb R^n$ be an $n$-dimensional Euclidean geometric algebra over $\mathbb R$, i.e. $e_i^2 = 1$ and $e_i\cdot e_j = 0$ for each $i \not= j$. Let $$\begin{gathered} \mathscr R(x; s) = \{x' \in \mathbb R^n \;:\; |(x'-x)\cdot e_i| < s/2,\; i = 1,\cdots, n\}, \\ \partial\mathscr R(x; s) = \bigoplus_{i=1}^n S^+_i(x; s)\oplus S^-_i(x; s),\quad S^\pm_i(x; s) = \{x' \in \mathbb R^n \;:\; (x'-x)\cdot e_i = \pm s/2\} \end{gathered}$$ which is a hypercube centered on $x$ with edge length $s \in (0, 1]$ and each face orthogonal to some $e_i$; the $\oplus$ in the expression for the boundary is to indicate that each component can be integrated separately, since their pair-wise intersections have no area.

We will take $\lim_{\mathscr R(x; s)\to 0} F(\mathscr R(x; s)) = L$ for multivectors $F(\mathscr R), L$ to mean that $$ \lim_{s\to0^+}||F(\mathscr R(x; s)) - L|| = 0,\quad ||X|| = \sqrt{\langle X\widetilde X\rangle} $$ where $\langle\cdot\rangle$ is the scalar part projection and $\widetilde X$ is the reverse of $X$. It doesn't matter what norm you choose since $\mathcal G_n$ is a finite-dimensional vector space, and I will only use the facts $$\begin{gathered} \lim F = L \implies \lim aFb = aLb, \qquad \lim F = L \implies \lim a{\rfloor}F = a{\rfloor}L \\ \lim(F + G) = \lim F + \lim G, \end{gathered}$$ for constant vectors $a, b$ and where in the addition formula all limits are assumed to exist. For the contraction formula, note that $$ a{\rfloor}F = \frac12(aF - \hat Fa),\quad \langle(\hat F -\hat L)(\hat F - \hat L)\widetilde{}\rangle = \langle(F - L)(F - L)\widetilde{}\rangle\widehat{} = \langle(F - L)(F - L)\widetilde{}\rangle, $$ where $\hat F$ is the grade involution of $F$.

Now let $f : \mathbb R^n \to \mathcal G_n$ be a function such that $$ \nabla f(x) = \lim_{s\to0^+}\frac1{|\mathscr R(x;s)|I}\int_{\partial\mathscr R(x;s)}\mathrm d^{n-1}x'\,f(x') $$ exists. Then $$\begin{aligned} \nabla f(x) &= \lim_{s\to0^+}\frac1{s^n}\int_{\partial\mathscr R(x;s)}|\mathrm d^{n-1}x'|\,\hat Nf(x') \\ &= \lim_{s\to0^+}\frac1{s^n}\sum_{i=1}^n\left[\int_{\partial S^+_i(x;s)}|\mathrm d^{n-1}x'|\,e_if(x') - \int_{\partial S^-_i(x;s)}|\mathrm d^{n-1}x'|\,e_if(x')\right] \\ &= \lim_{s\to0^+}\frac1{s^n}\sum_{i=1}^ne_i\left[\int_{\partial S^+_i(x;s)}|\mathrm d^{n-1}x'|\,f(x') - \int_{\partial S^+_i(x;s)}|\mathrm d^{n-1}x'|\,f(x'-se_i)\right] \\ &= \lim_{s\to0^+}\frac1{s^n}\sum_{i=1}^ne_i\int_{\partial S^+_i(x;s)}|\mathrm d^{n-1}x'|\,\bigl(f(x') - f(x'-se_i)\bigr) \end{aligned}$$ Applying $e_k{\rfloor}$, we see that each limit within the sum exists, so $$ \nabla f(x) = \sum_{i=1}^ne_i\lim_{s\to0^+}\frac1{s^n}\int_{\partial S^+_i(x;s)}|\mathrm d^{n-1}x'|\,\bigl(f(x') - f(x'-se_i)\bigr). \tag{$**$} $$ Let $J_i(x; s)$ be the expression inside the limit on the last line and $F_i(y; s) = f(y) - f(y-se_i)$. Then $$\begin{aligned} ||J_i(x; s) - f_i(x)|| &\leq ||J_i(x; s) - \frac1s F_i(x; s)|| + ||\frac1s F_i(x; s) - f_i(x)|| \\ &= ||J_i(x; s) - \frac1s F_i(x; s)|| + o(1), \end{aligned}$$ where $f_i(x) = \partial f(x)/\partial x_i$. The first term can be bounded as $$\begin{aligned} ||J_i(x; s) - \frac1s F_i(x; s)|| &= ||\frac1{s^n}\int_{S^+_s(x;s)}|\mathrm d^{n-1} x'|\bigl(F_i(x'; s) - F_i(x; s)\bigr)|| \\ &\leq \frac1s\sup_{x'\in S^+_i(x;s)}||F_i(x'; s) - F_i(x; s)|| \end{aligned}$$ Inside the supremum, $$\begin{aligned} ||F_i(x'; s) - F_i(x; s)|| &= ||f(x') - f(x) + f(x - se_i) - f(x' - se_i)|| \\ &= ||Df(x)[x'-x] + Df(x)[x - x'] + o(|x' - x|)|| \\ &= o(|x' - x|), \end{aligned}$$ where $Df(x) : \mathbb R^n \to \mathbb R^n$ is the total derivative of $f$ at $x$ and $o(\cdot)$ is little $o$ notation. Since the furthest $x'$ can be from $x$ is the corner of the cube, we get $$ |x' - x| \leq \frac s2\sum_{i=1}^ne_i = \frac{s\sqrt n}2, $$ so $||F_i(x'; s) - F_i(x; s)|| = o(s)$, and altogether $||J_i(x; s) - f_i(x)|| = o(1) \to 0$ as $s \to 0^+$. Hence, returning to ($**$) we finally have $$ \nabla f(x) = \sum_{i=1}^n e_if_i(x) = \sum_{i=1}^n e_i\frac{\partial f}{\partial x_i}(x). $$

[1] C. Doran, A. Lasenby, Geometric Algebra for Physicists, Cambridge University Press, Cambridge, 2003.

[2] L. Dorst, The Inner Products of Geometric Algebra, in: L. Dorst, C. Doran, J. Lasenby (Eds.), Applications of Geometric Algebra in Computer Science and Engineering, Birkhäuser, Boston, 2002: pp. 35–46.

[3] D. Hestenes, G. Sobczyk, Clifford algebra to geometric calculus: a unified language for mathematics and physics, D. Reidel, Dordrecht, 1984.

[4] A. Macdonald, Sobczyk’s simplicial calculus does not have a proper foundation, arXiv:1710.08274 [Math]. (2017).

[5] G. Sobczyk, O.L. Sánchez, Fundamental Theorem of Calculus, Adv. Appl. Clifford Algebras. 21 (2011) 221–231.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .