HELP! A Stubborn Vector Identity to UnderstandPosted: February 23, 2016
Over the past three years or so, I have been researching the history and implementation of Gibbsian vector analysis with the intent of finding ways to incorporate it more thoroughly and more meaningfully into introductory calculus-based physics (possibly algebra/trig-based physics too). Understanding the usual list of vector identities has been part of this research. One vector identity that has frustrated me involves probably the most innocent looking quantity, the gradient of the dot product of two vectors. I have seen no fewer than five different expressions for the expansion of this seemingly harmless quantity. Here they are.
Now, equation (1) uses Feynman notation, which endows the nabla operator with the property of obeying the Leibniz rule, or the product rule, for derivatives. The subscript refers to the vector on which the nabla operator is operating while the other vector is treated as a constant. Note that in chapter 3 of Wilson’s text based on Gibbs’ lecture notes, the subscript denotes which vector is to be held constant, precisely the opposite of the way Feynman presents it. Equation (1) is merely an alternative way of writing the lefthand side and offers nothing new algebraically.
Equation (2) shows nabla operating on each vector in the dot product, which is something many students never see. Like I was told years ago, they are told that one can only take the gradient of a scalar and not a vector, which is patently false. The twist is that, unlike the gradient of a scalar, the gradient of a vector is not a vector; it is a second rank tensor which can be represented by a matrix. This tensor, and its matrix representation, is also called the Jacobian. The dot product of this tensor with a vector gives a vector, so equation (2) is consistent with the fact that the lefthand side must be a vector. I can derive this expression using index notation.
Equation (3) is equation (2) written in (a very slight variation of) matrix notation (the vectors are written as vectors and not as column matrices). I don’t think there is anything more to it.
Equation (4) is the traditional expansion of the lefthand side. It is derived from the BAC-CAB rule, with suitable rearrangements to make sure nabla operates on one vector in each term. Two such applications give equation (4). The “reverse divergence” operators are actually directional derivatives operating on the vectors immediately to the right of each operator. I can derive this expression using index notation.
Equation (5) is shown in problem 1.8.12 on page 48 of Arfken (6th edition). It has the advantage of using the divergences of the two vectors, which I think are easier to understand than the “reverse divergence” operators in equation (4). However, the “reverse curl” operators are completely new to me and I have never seen them in the literature anywhere other than in this problem in Arfken. I think this equation can be derived from equation (4) by appropriately manipulating the various dot and cross products. I have not yet attempted to derive this expression with index notation.
Now, many questions come to mind. I have arranged the first and second terms on the righthand sides of equations (4) and (5) to correspond to the first term on the righthand sides of equations (1), (2), and (3). Similarly, the third and fourth terms on the righthand sides of (4) and (5) correspond to the second term on the righthand sides of equations (1), (2), and (3). By comparison, this must mean that somehow from the gradient (Jacobian) of a vector come both a dot product and a triple cross product. How can this be?
How can the gradient (Jacobian) of a vector be decomposed into a dot product and a triple cross product?
I think I can partly see where the dot product comes from, and it’s basically the notion of a directional derivative. The triple cross products are a complete mystery to me. Is there a geometrical reason for their presence? Would expressing all this in the language of differential forms help? Equations (4) and (5) also seem to imply that the triple cross products are associative, which they generally are not. I think I can justify the steps to get from (4) to (5), so if anyone can help me understand geometrically how the Jacobian can be decomposed into a dot product (directional derivative) and the cross product of a vector and the curl of the other vector, I’d be very grateful.