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The origin of the “nabla” symbol

February 20, 2012, 7:32 pm

We’re about to start working with gradient vectors in Calculus 3, and this topic uses a curious mathematical symbol: the nabla, which looks like: \(\nabla\). This symbol has several mathematical uses, one of which is for gradients; if \( f  \) is a function of two or more variables then \( \nabla f \) is its gradient. But there does not appear to be a use for the symbol outside mathematics (and mathematical physics).

One of my students asked me about the origin of this symbol, and I had to confess I didn’t know. I always figured it was somehow related to the much more common capital Greek delta, \( \Delta \), but the real story is a lot more colorful than that.

The nabla is so-called because it looks like a harp; the Greek word for the Hebrew or Egyptian form of a harp is “nabla” . What does a harp have to do with mathematics? The image came up in relation to mathematics through part of a conversation between Peter Guthrie Tait and James Clerk Maxwell:

The name was used in playful intercourse between Tait and Clerk Maxwell, who in a letter of uncertain date finished a brief sketch of a particular problem in orthogonal surfaces by the remark “It is neater and perhaps wiser to compose a nablody on this theme which is well suited for this species of composition.” [...]

It was probably this reluctance on the part of Maxwell to use the term Nabla in serious writings which prevented Tait from introducing the word earlier than he did. The one published use of the word by Maxwell is in the title to his homorous Tyndallic Ode, which is dedicated to the “Chief Musician upon Nabla,” that is, Tait.

In a letter from Maxwell to Tait on Nov. 7, 1870, Maxwell wrote, ”What do you call this? Atled?”

In a letter from Maxwell to Tait on Jan. 23, 1871, Maxwell began with, ”Still harping on that Nabla?”

(The above quote is from this text file that contains a lot more on this subject.)

I like that term “atled” — that’s the backwards spelling of “delta” of course. I keep thinking I’ve seen “atled” as an alternative \( \LaTeX \) command for nabla.

It’s also nice to see two giants in mathematics have a bit of fun together as they developed some of their mathematics. The linked article mentions that other mathematicians apparently more stuffy than Tait and Maxwell, were reluctant to adopt this symbol and its name because it didn’t seem serious enough. The joke’s on them!

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