# Notations for Laplacian: $\nabla^2$ vs. $\Delta$

For a (sufficiently smooth) function $$f\colon \Bbb R^n\to\Bbb R$$, the Laplacian of $$f$$ is defined to be $$\sum_{j=1}^n \frac{\partial^2 f}{\partial x_j^2}$$. There are two notations for the Laplacian that I have seen being commonly used, viz., $$\nabla^2 f$$ and $$\Delta f$$. The first one is mostly used by physicists, whereas mathematicians tend to prefer the latter one (at least in my area of study, where $$\nabla^2 f$$ is reserved for the Hessian of $$f$$).

I would like to ask if anyone knows the origin of these two usages, in particular regarding the people who first introduced the symbols and the people who popularized them.

What follows is from A History of Vector Analysis by Michael John Crowe (1967; 1985 Dover corrected reprint).

(from middle of p. 167) In Heaviside's later papers of 1883 and 1884 use was made of vectors, but no new principles were $$\text{introduced.}^{35}$$

(from middle of p. 180) $${}^{35}$$This should perhaps be qualified by the statement that [the symbol] $$\nabla^2$$ made its first appearance in an 1884 paper. See (5,I; 338). No comment was given by Heaviside about its meaning. He seems to use it as $$+\left(\frac{d^2}{dx^2} + \frac{d^2}{dy^2} + \frac{d^2}{dz^2}\right)$$ rather than, as Maxwell and Tait had done, as the negative of the above.

I have no idea what specific reference (paper/book title) Crowe's cryptic code refers to. There is no bibliography (by chapter or for the entire book). After several minutes of looking through the book (preface, beginning of chapter notes, various things at end of the book, etc.), I gave up trying to figure out what it means. I suspect the reference is to Volume I of Heaviside's collected works, and this is probably mentioned somewhere in pages before this, but I don't have time now to keep looking. (rant) I think there's an important lesson to be learned here for anyone wishing to write something like Crowe's book: Don't use your own private bibliographic code unless you clearly indicate how to decipher it, in a location that a casual user can reasonably be expected to find without much difficulty.

I think the reason some people used a negative sign is because of the influence of quaternions, where squares of i, j, k are negative.

Possibly useful to look over is Vector Analysis by Gibbs/Wilson (1901), which I believe is the primary text that popularized and spread vector ideas beyond the relatively few researchers that had up to that time been using them.

Finally, see p. 2 of Peter Guthrie Tait, [untitled address to the Mathematics and Physics section], pp. 1-8 in the 2nd paging of Report of the Forty-First Meeting of the British Association for the Advancement of Science (Edinburgh, 2−9 August 1871), John Murray (London), 1872, cv + 207 + 281 + iv + 83 pages. (If anyone is curious as to how I happen to know about Tait's comments, see these 2 comments.)

• Thank you for the answer! I really appreciate your effort of trying to decipher the bibliographic code even though it wasn't very successful. Sep 12 '21 at 13:31
• It's weird that Crowe doesn't include the vector component symbols $\vec{i}$, $\vec{j}$, $\vec{k}$ in his representation of Heaviside's meaning. Sep 12 '21 at 14:34
• @Spencer: Unit vectors are not needed, since $\nabla^2$ is a scalar. As for the use of unit vectors in $\nabla$ and in $\nabla \cdot \nabla$ (formal dot product), this appears in several places (e.g. p. 131). From bottom of p. 136: >Thus in Cartesian analysis $\nabla^2$ can be defined as either plus or minus $\left(\frac{d^2}{dx^2} + \frac{d^2}{dy^2} + \frac{d^2}{dz^2}\right).$ [Lord] Kelvin used the positive sign, whereas Maxwell used the negative sign and noted: "The negative sign is employed here in order to make our expressions consistent with those in which Quaternions are employed." Sep 12 '21 at 15:06
• (5,I;338) means Heaviside's Electrical Papers, London, 1892 (referenced in Note 5 to Crowe's chapter), volume I, p. 338. This is probably a typo, there is no $\nabla^2$ on p. 338, but there is one on p. 358 in the equation $\nabla^2H=\frac{4\pi\mu}{\rho}\dot{H}$ for pure conductors. Published as The Induction of Currents in Cores in The Electrician, p. 583ff, May 3, 1884. Sep 13 '21 at 4:41
• @Conifold: The '5' as referring to "Note 5" in the same chapter was one of the things I thought might be intended (based on "Concerning bibliography" paragraph in the Preface), but the fact that nothing relevant was on p. 338 led me to doubt that meaning. Yesterday I returned to this issue for a few minutes and tried looking in Macfarlane's bibliography, but "5,I; 338" didn't lead me to anything. I'll update the answer to include the reference when I get a chance later. Sep 13 '21 at 12:33

Jeff Miller's site gives the first occurrence of $$\Delta$$ as

The symbol $$\Delta$$ for the Laplacian operator (also represented by $$\nabla^2$$) was introduced by Robert Murphy in 1833 in Elementary Principles of the Theories of Electricity. (Kline, page 786)

The first use of the term Laplace's Operator is given as

The term LAPLACE’S OPERATOR (for the differential operator $$\nabla^2$$) was used in 1873 by James Clerk Maxwell in A Treatise on Electricity and Magnetism (p. 29): "...an operator occurring in all parts of Physics, which we may refer to as Laplace’s Operator" (OED).

The first use by physicist of the term Laplacian for the $$\nabla^2$$ operator is given on the same page as:

LAPLACIAN (as a noun, for the differential operator $$\nabla^2$$) was used in 1935 by Pauling and Wilson in Introd. Quantum Mech. (OED).

Unfortunately, these entries do not make clear when the $$\nabla^2$$ notation was first introduced.

• This is very informative, thanks! Sep 12 '21 at 13:32

I believe the notation $$\nabla^2 f$$ comes from $$\nabla^2 f = \nabla\cdot\nabla f = \operatorname{div}(\operatorname{grad} f)$$ See here

• Do you happen to know who was the first mathematician/scientist to use $\nabla^2$ to represent $\nabla \cdot \nabla$? It's a very poor choice of notation in my opinion. Sep 11 '21 at 16:31
• @BigbearZzz: $\nabla^2$ was a very natural choice of notation for the times. Keep in mind that we're talking about mid to late 1800s English works (mostly), and at the time the Calculus of Operations and related "algebrizations" (e.g. Cayley's quantics) was in fashion in Great Britain. For example, see The calculus of operations and the rise of abstract algebra by Elaine H. Koppelman (1971), especially p. 238. Sep 12 '21 at 15:25
• @DaveLRenfro It still seems weird at the very least since one would expect the mixed terms like $\frac{\partial^2}{\partial x\partial y}$ to be presented in $\nabla^2$. Using $|\nabla|^2$ would make more sense I think (unless the symbol $|v|$ for vector norm didn't exist back then). Sep 12 '21 at 17:08
• @BigbearZzz: If you look through the 1901 Gibbs/Wilson book, you'll find the dot product version almost exclusively used. As for using $|\nabla|^2,$ I guess even if the norm symbol was in use then (and I don't think it was), $\nabla^2$ would probably be preferred simply to avoid clutter and because the context would be clear and I don't think $\nabla^2$ was manipulated algebraically in any nontrivial way. Finally, at least with quaternions you don't have cross terms. The quaternion product $(a\text{i}+b\text{j}+c\text{k})(a\text{i}+b\text{j}+c\text{k})$ is equal to $-(a^2+b^2+c^2).$ Sep 12 '21 at 19:59
• @DaveLRenfro The quaternion product analogy makes so much sense, I have never thought about it that way before. Thank you very much. Sep 13 '21 at 1:26