Conceptual Understanding in Introductory Physics XXVIII

You may not agree that the topic(s) of this question belong in an introductory calculus-based physics course, but I’m going to pretend they do for the duration of this post. Gradient, divergence, and curl are broached in Matter & Interactions within the context of electromagnetic fields. Actually, gradient appears in the mechanics portion of the course.

One problem with these three concepts, especially divergence and curl, is the distinction between their actual definitions and how they are calculuated. The former are rarely, if ever, seen at the introductory level and usually first appear in upper level courses. However, some authors [cite examples here] replace the physical definitions with the mathematical symbols invented by Heaviside and Gibbs to represent the calculation of these quantities. In other words, the divergence of \mathbf{A} is frequently defined as \nabla\cdot\mathbf{A} and the curl of \mathbf{A} is frequently defined as \nabla\times\mathbf{A}. These should be treated as nothing more than symbols representing their respective physical quantities and should not be taken as equations for calculation. If one insists on keeping this notation, then the dot and cross should at least be kept with the nabla symbol so that \nabla\cdot represents divergence and \nabla\times represents curl. Either way, these are operators that operate on vectors and their symbols should reflect that concept and should be interpreted as such and not as a recipe for calculation. This book by Tai was extremely helpful in getting this point across to me.

Gradient has its own unique problem in that some sources claim that one can only take the gradient of a scalar, which is patently false. One can indeed take the gradient of, for example, a gradient but the object one gets back is not a vector. If we adopt a unified approach to vector algebra and vector calculus we find that there are patterns associating the operand and the result when using these vector opators. For example, operating on a vector with \nabla doesn’t produce a vector; it produces a second rank tensor. This is one reason I would love to find a way to bring this approach into the introductory course. So many things would be unified.

But now, on to the questions I want to ask here

(a) Write a conceptual definition of gradient in words.

(b) Write a mathematical definition of gradient that does not depend on any particular coordinate system. You must not use the nabla symbol.

(c) Write a conceptual definition of divergence in words.

(d) Write a mathematical definition of divergence that does not depend on any particular coordinate system. You must not use the nabla symbol.

(e) Write a conceptual definition of curl in words.

(f) Write a mathematical definition of curl that does not depend on any particular coordinate system. You must not use the nabla symbol.

Go!

(Note: I need to revisit this post in the future to make sure the notion of applying gradient to a vector quantity can be handled in the coordinate free way I have in mind. My intuition is that it can be, but I need to work out some details. )

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