How were vector calculus nabla ∇ identities first derived?

Question

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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?

Answer 1

Index notation is just a shorthand for coordinate computations, which becomes necessary in many dimensions, but is merely convenient in dimensions 3 and 4. There is no need to even index variables there, they can just be denoted by different letters.

You can see how Hamilton dealt with nabla identities in sections 49-50 of his seminal 1844 paper On Quaternions. Basically, he expresses them in coordinates and manipulates coordinate expressions according to the rules of quaternion algebra. Maxwell does the same in part VI of his Dynamical Theory of Electromagnetic Field, except without quaternions, he derives formulas in coordinates and then rewrites them with nabla. By the way, Maxwell was the one who called nabla "nabla" in jest, by the name of similarly shaped Egyptian harp. Gauss and Green formulas were originally expressed, and derived, without index notation, or even nabla.

The big advantage of Gibbs's symbolic vector calculus, which appeared in draft before 1888 and was systematized in his 1901 book with Wilson, was that he listed the basic identities and offered rules by which more complicated ones could be derived from them. His formalism was incomplete however, some identities do not reduce to basic ones and have to be derived from coordinates directly, and it only worked in 3 dimensions. So Einstein couldn't use it in general relativity. Tensor formalism was developed by Ricci and Levi-Civita in 1890-s, and they are the ones who championed the index notation, see more here. Einstein appropriated it, and introduced some simplifying conventions, including the summation rule.

To get a formalism completely free of coordinates, that subsumes vector calculus and generalizes to higher dimensions, one needs Cartan's differential forms, the differential, flats, sharps and Hodge star. It was not developed until 1950-s.