Discussion between Michael Boratko and Paul

Stack Exchange

02:42

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 oriente...

Michael Boratko

What's the source of this notation? I have not seen it written this way.

Paul

Its in Understanding and Implementing the Finite Element Method by M.S. Gockenbach.

Michael Boratko

Ah, yes I have seen it written that way before, I was misinterpreting it.

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.

Paul

@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?

Michael Boratko

Again, I'm hesitant to say yes unequivocally because I have not used the notation myself often. That being said, I would think that $dS_1$ would be the same as $ds$ in your typical line integral notation, so yes I believe you are correct.

Paul

02:42

When I evaluate it this way, the integral does not come out to the same value as predicted by green's theorem.

Michael Boratko

@Paul Is the integrand a vector field or a scalar field?

In particular, how are you applying Green's theorem if it is a scalar field?

A line integral whose differential is $ds$ is the line integral of a scalar field

(with respect to arc length)

Green's Theorem applies to vector fields in $\mathbb R ^2$

the integrand in question looks like a scalar field

@Paul Can you post the integrand?

Paul

The integrand is a scalar quantity $F\cdot n$, where $F=v\nabla u$

Michael Boratko

right

so how were you applying Green's Theorem?

Paul

I'm verifying Green's theorem by explicitly evaluating the line integral.

for a specific u, v

Michael Boratko

Yes, but Green's theorem doesn't apply here

Green's theorem

In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. It is named after George Green, and is the two-dimensional special case of the more general Stokes' theorem. Theorem Let C be a positively oriented, piecewise smooth, simple closed curve in the plane \mathbb{R} 2, and let D be the region bounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there, then :\oint_{C} (L\, \mathrm{d}x + M\, \mathrm{d}...

Paul

02:51

Sorry... I should have said: Divergence theorem.

Michael Boratko