# How were vector calculus nabla ∇ identities first derived?

(Math Stack Exchange suggested that the same question I posted there be migrated here; The one at Math Stack Exchange was thus deleted. The recommendation message of migration can be found here, though the page is now deleted.)

Whenever students are asked to derive or prove some vector calculus identities such as

$$\nabla^2\vec{A}=\nabla(\nabla\cdot \vec{A})-\nabla \times (\nabla \times \vec{A})$$

they are often asked to expand in index notation and rearrange to give the required expressions.

This caused me to wonder how these identities were first derived (since not all of them are consequence of the product rule of derivatives).

A brief search of the history of the topic seemed to suggest the following:

Complex numbers $\to$ quaternions $\to$ Grassmann (exterior algebra) $\to$ Vector analysis.

Another brief search on the topic of tensors and Albert Einstein showed that the two periods of development of vector analysis and tensor analysis have some overlap thus it might be possible they might be aware of the development of each other.

However, since the vector analysis history account above does not mentioned about tensors nor index notation, how were the vector calculus identities originally derived, possibly without index notation available at that time?

• This is not the "exterior algebra" but "exterior calculus" (calculus of differential forms) which was introduced after vector calculus (by E. Cartan in 1920-th). Feb 8 '15 at 13:32

• @Javier Hamilton introduced the symbol, but he had no name for it and wrote it sideways. Maxwell rotated it by $90^\circ$ as we write today, so it looks more like the harp, and nicknamed it "nabla", Gibbs adopted Maxwell's transcription but not the name, his version was "del" because it is "so short and easy to pronounce that even in complicated formulae in which $\nabla$ occurs a number of times, no inconvenience to the speaker or listener arise's". Feb 9 '15 at 17:16