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Vector calculus, usually used in Physics etc., is the branch of mathematics that deals with calculus operations in three-dimensional Euclidean space.

The scalar Laplacian operator (is usually denoted by $\nabla^2$ or $\nabla \cdot \nabla$, where $\cdot$ is the dot product) of a potential function $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ is given by

$$ \nabla^2 f = \frac{\partial^2 f}{\partial x_1^2} + \frac{\partial^2 f}{\partial x_2^2} + \frac{\partial^2 f}{\partial x_3^2}, $$ where $\mathbf{x} = (x_1, x_2, x_2)$ is the input vector of $f$.

The same notation is used for the so-called vector Lapacian operator, which is applied on a vector field, say, $\mathbf{F}: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ given by

$$ \nabla^2 \mathbf{F} = \left(\frac{\partial^2 F_1}{\partial x_1^2}, \frac{\partial^2 F_2}{\partial x_2^2}, \frac{\partial^2 F_3}{\partial x_3^2} \right), $$ where $\mathbf{F} = F_1 \hat{\mathbf{i}} + F_2 \hat{\mathbf{j}} + F_3 \hat{\mathbf{k}} $

So far, so good, I can implicitly differ scalar from vector Laplacian operator them by observing whether the operator $\nabla^2$ is being applied to a potential (scalar-valued) function or a vector field (vector-valued) function.

The problem is, sadly, it becomes ambiguous when you also use the notations of Matrix Calculus, which is another branch of mathematics that commonly deals with higher order dimensions, being therefore used in many fields, such as Convex Optimization, Statistical Signal Processing, Machine Learning, etc.

For instance, in denominator layout, the Hessian matrix is usually denoted as (take the Simon Haykin book as reference, eqs. (3.15) and (4.54)) $$ \mathbf{H} = \nabla^2 f = \dfrac{\partial^{2} f(\mathbf{x})}{\partial \mathbf{x}^2} = \left[ \begin{matrix} \dfrac{\partial^{2} f}{\partial x_1^2} & \dfrac{\partial^{2} f}{\partial x_1 \partial x_2} & \cdots & \dfrac{\partial^{2} f}{\partial x_1 \partial x_n} \\ \dfrac{\partial^{2} f}{\partial x_2 \partial x_1} & \dfrac{\partial^{2} f}{\partial x_2^2} & \cdots & \dfrac{\partial^{2} f}{\partial x_2 \partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac{\partial^{2} f}{\partial x_n \partial x_1} & \dfrac{\partial^{2} f}{\partial x_n \partial x_2} & \dots & \dfrac{\partial^{2} f}{\partial x_n^2} \end{matrix} \right] $$

The very same notation $\nabla^2$ is used. The mathematical computation, however, is completely different. Note that, for the gradient, $\nabla f$, both branches of mathematics agree as their results are the same.

My question is: how should I deal with this ambiguity in a situation where the application is based on both branches of mathematics?

Rubem Pacelli
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    The cheapest way is to write the Laplacian as $\Delta,.$ This is quite common. – Kurt G. May 05 '23 at 18:45
  • @KurtG. It is overused for the coefficient parameter update, though :( – Rubem Pacelli May 05 '23 at 18:47
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    You won't find an authority that gives you an unambiguous recommendation for notation. Notation is opinion based. My personal rules are: it has to help the reader to understand what I am doing. And if the latter is not obvious I have to explain this as necessary. – Kurt G. May 05 '23 at 18:49
  • You could use this so the three expressions' entries are $\partial_i\partial_if,,\partial_i\partial_iF_j,,\partial_i\partial_jf$ or, in a more concise notation, $f_{,ii},,F_{j,ii},,f_{,ij}$ (some people would argue those each need two commas). – J.G. May 05 '23 at 18:52
  • $D^2f$ is far more common for the Hessian. – Ted Shifrin May 05 '23 at 19:19
  • @TedShifrin which authors use it? – Rubem Pacelli May 05 '23 at 19:28
  • I am in no position to list names, sorry. Pretty much every analysis textbook I’ve seen, for starters. – Ted Shifrin May 05 '23 at 19:33
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    I don't agree with your vector Laplacian, it should be imho $\nabla^2 \mathbf{F} = \frac{\partial^2 \mathbf{F}}{\partial x_1^2}+ \frac{\partial^2 \mathbf{F}}{\partial x_2^2}+ \frac{\partial^2 \mathbf{F}}{\partial x_3^2} $ – Vincenzo Tibullo May 05 '23 at 19:50
  • @VincenzoTibullo That is named "scalar Laplacian operator" as it results in a scalar. The vector Laplacian results in a vector. For more info, see here – Rubem Pacelli May 05 '23 at 20:00
  • What @VincenzoTibullo just wrote does not give a scalar, actually, ... – paul garrett May 05 '23 at 20:19
  • @paulgarrett Yes. Thank you for pointing out. The correct form of the scalar Laplacian operator is $\nabla^2 f = \frac{\partial^2 f}{\partial x_1^2} + \frac{\partial^2 f}{\partial x_2^2} + \frac{\partial^2 f}{\partial x_3^2}$, where $f$ is a scalar-valued function, just as I wrote in this post – Rubem Pacelli May 05 '23 at 22:36
  • @paulgarrett personally, I never saw the definition of Vincenzo Tibullo . Nevertheless, my definition of vector Laplacian agrees with wikipedia. – Rubem Pacelli May 05 '23 at 22:39
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    From Wikipedia: "When computed in orthonormal Cartesian coordinates, the returned vector field is equal to the vector field of the scalar Laplacian applied to each vector component." it means $$\nabla^2\mathbf{F}=(\nabla^2F_1,\nabla^2F_2,\nabla^2F_3)\neq\left(\frac{\partial^2\mathbf{F}}{\partial x_1^2},\frac{\partial^2\mathbf{F}}{\partial x_2^2},\frac{\partial^2\mathbf{F}}{\partial x_3^2}\right)$$ – Vincenzo Tibullo May 06 '23 at 06:24
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    I've often seen $\def\n{\nabla} \n^2$ used to denote the Laplacian (and occasionally $\Delta$). Notation for the Hessian is less standardized and I've seen all of the following: $,\n^2,,\n\n^T,,\n\otimes\n,,\n\n,,D,,H.;$ The least ambiguous notation IMHO is to employ parentheses, i.e. $,\n(\n f)$ – greg May 06 '23 at 15:05
  • @VincenzoTibullo I apologize for that mistake. What you wrote was exactly what I meant to say. I was very tired when I made this post, actually hahaha Now I suppose that is right. Once again, the definition that Vincenzo Tibullo wrote is totally different from what we see on Wiki. – Rubem Pacelli May 06 '23 at 21:36
  • I agree with @greg, the notation $\nabla (\nabla f)$ seems the unambiguous way the denote the Hessian matrix. – Rubem Pacelli May 11 '23 at 17:47

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