In section section 27-3 of The Feynman Lectures on Physics, Feynman describes a notation for manipulating vector expressions in a way that endows nabla with the property of following a rule similar to the product rule with which our introductory calculus students are familiar. It allows a vector expression with more than one variable to be expanded as though nabla operates on one variable while the other is held constant. The vector being differentiated is indicated with a subscript on nabla. Feynman’s equation 27.10 shows how this is written, and it rather like treating the subscripted nablas as partial derivative operators. Feynman’s equation 27.11 shows the resulting vector identity for the divergence of a cross product. In between these two equations. Feynman explains that the subscripted nabla can be manipulated as though it were a vector (it is not) according to the rules of dot products (commutative), cross products (anticommutative), triple scalar products (cyclic permutation, swapping dots and crosses, etc.), and triple vector products (BAC-CAB, Jacobi identity, etc.) The strategy is to end up with only one vector (the one corresponding to a subscript) immediately to the right of each correspondingly subscripted nabla. Then you drop the subscripts, and you should have a valid vector identity. In the audio version of this lecture, Feynman comments that he doesn’t understand why this technique isn’t taught. It was never shown to me as either an undergraduate or graduate student. I suspect it’s treated as “one of those things” students are simply assumed to pick up at one point or another without it ever being explicitly addressed (much like critical thinking is treated).
The issue here, for me, is whether or not Feynman invented this way of manipulating vector expressions. After all, the notation carries his name so it might be reasonable to assume he invented the underlying method. My research shows that a very similar methodology is documented in the very first (as far as I know) textbook on vector analysis, Wilson’s Vector Analysis: A Text-Book for the use of Students of Mathematics and Physics. This is the famous work based on Gibbs’ lecture notes and is the definitive work on contemporary vector analysis. I continue to be surprised at how few people have consulted it (based on my asking whether or not they have). I offer the PDF version to my physics students in the hopes they will use it in their studies. Chapter 3 is on the differential calculus of vectors and section 74 on page 159 begins a presentation of using nabla as a “partial” operator in an expression, operating on only one vector while holding another constant. Wilson introduces a subscript notation that, unlike Feynman’s, indicates which vector is held constant for a differentiation.
This brings to my mind the question of whether or not Feynman was aware of Wilson’s textbook and this method documented therein and decided to change the nature of the subscript to show what is differentiated rather than what is not. I don’t see how there is any way to know for sure, but it’s an interesting question in my mind because I suspect many students are not aware of Wilson’s textbook.
Wilson shows many worked examples on subsequent pages. Section 75 on page 161 shows more examples and consequences of this technique leading to a statement on page 162 that blows my mind! In the paragraph immediately surrounding equation (47) we see the following:
If u be a unit vector, say a, the formula (referring to equation 47) expresses the fact that the directional derivative (expression omitted) of a vector function v in the direction a is equal to the derivative of the projection of the vector v in that direction plus the vector product of the curl of v into the direction a.
Wow! This mean that applying nabla as a partial operator leads to something of geometrical significance, which to me constitutes a new identity itself. The lefthand side of Wilson’s equation (47) can be interpreted as the dot product of vector a and the gradient of vector v (a second rank tensor). My last post asks how the righthand side follows geometrically from that, something I’ve never seen in the literature.
Tai’s recent book on vector and dyadic analysis presents what he calls the “method of symbolic vector” which seems to formalize Wilson’s both Feynman’s methods. The idea is that nabla is temporarily treated as a vector (with a new symbol) and any expression in which is appears can be symbolically manipulated according to all the rules of vector analysis to end up with a valid identity when nabla is once again treated as a differential operator (and restored to its rightful symbol). Tai definitely knew about Wilson’s text as he references it frequently and devotes a considerable number of pages to commentary on Gibbs’ choices of notation (e.g. Gibbs’ use of a dot product as a symbol for divergence despite divergence not being defined as a dot product at all, and similarly his use of a cross product as a symbol for curl despite curl not being defined as a cross product), etc. Tai refers to Feynman only once, at the bottom of page 147 and continuing onto page 148, but the reference is vague.
Regardless of who initially invented the use of nabla as a partial operator, I feel we need to expose students to this as early as possible as part of a stronger foundation in classical vector analysis than they currently get in the introductory courses.