How Does the Divergence Operator Apply in the Transport Theorem?

[victor.raum]

I'm reading about the transport theorem in my vector calculus book. They state the following at the beginning of the section:

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Let F be a vector field on R^3. Let c(x, t) denote a flow line on F starting at location x and continuing out for t seconds. Let J(x, t) denote the Jacobian of c with respect to x, and with t fixed. We then have

dJ/dt = [div F(c(x, t))] * J

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Now, what confuses me is the [div F(c(x, t))] part. Does that mean the divergence of F evaluated at c, or does it mean the divergence of of the composite function (F o c)? In other words, how tightly does the divergence operator bind to its operand?

Expanding out dJ/dt piece by piece tells me that the statement in question should read "the divergence of F evaluated at c." I am pretty sure I did the expansion correctly, but I would still be curious to hear someone else chime in.

P.S. Apologies for not TeXing any of this post. I only know Plain TeX, but my understanding is that the forums here only accept LaTeX.

 

[eljose79]

Thank you for your question about the transport theorem and the interpretation of the divergence operator in this context. You are correct in your understanding that the statement should read "the divergence of F evaluated at c." This means that the divergence operator is applied to the vector field F at the point along the flow line c, rather than to the composite function (F o c).

In general, the divergence operator is a mathematical operation that takes a vector field as its input and produces a scalar value as its output. The result of the operation is a measure of how much the vector field is "spreading out" or "diverging" at a particular point. In the context of the transport theorem, we are interested in how the flow line c affects the divergence of the vector field F at that point.

I hope this clarifies your understanding of the transport theorem and the role of the divergence operator in this context. If you have any further questions, please don't hesitate to ask. Thank you for your interest in vector calculus.