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I'm evaluating a line integral written in the form:

$\int_{\partial\Omega_1} v\nabla u\cdot n$ where $\partial \Omega_1$ is simple curve forming one part of the boundary $\partial\Omega$ of a closed region, and $n$ is the unit normal to $\partial\Omega_1$.

Suppose C(t) is a positively oriented parameterization of $\Omega_1$. Since there is no differential explicitly indicated in the integral, can i automatically assume that the differential is $||C'(t)||dt$?

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What's the source of this notation? I have not seen it written this way. – process91 Sep 14 '12 at 1:58
Its in Understanding and Implementing the Finite Element Method by M.S. Gockenbach. – Paul Sep 14 '12 at 2:04
Ah, yes I have seen it written that way before, I was misinterpreting it. – process91 Sep 14 '12 at 2:10
I believe you can. Compare, for instance, the notation in your book for the divergence theorem and this notation of the divergence theorem. Perhaps someone more familiar with the book will come along with a more definitive answer, however. – process91 Sep 14 '12 at 2:19
@MichaelBoratko: Yes, in fact, I'm evaluating the RHS integral in this notation. I'm parameterizing a curve around a quadrilateral, one segment at a time. Would $dS_{1}$ (in my case) be equivalent to ||C'(t)||dt? – Paul Sep 14 '12 at 2:24

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