I have an embarrassingly simple question about line integrals, if anyone is available.
The question in the book asks to compute the line integral with respect to arc length of $\int_C (x+y)ds$ where $C$ is a triangle with vertices $(0,0)$, $(0,1)$, and $(0,1)$ traversed in a counterclockwise direction
My book defines line integrals with respect to arc length as follows - if $f$ is a scalar field then the line integral of $f$ with respect to arc length along $C$ (parametrized as $g(t)$) is given by $\int_a^b f(g(t))s'(t)dt$, where $s'(t)=\lVert g'(t)\rVert$
The line integral of a vector function $\vec{f}$ along a path $\mathcal{P}$ is denoted by $\int_\mathcal{P} \vec{f} \cdot \mathrm{d} \vec{s}$ where $\mathrm{d}\vec{s}$ is a small line element.
@KannappanSampath The line integral with respect to arc length is just like the line integral of a vector function, however "the vectors [involved] are always tangential to the line".
(I should mention that I made a mistake above, I should have written $s_i(t)=\int \lVert g_i'(t) \rVert dt$, otherwise $s_i'(t)=\lVert g_i'(t)\rVert$ doesn't make much sense :)
@robjohn My book (Apostol's Calculus) defines general line integrals first, and then they go on to say that a line integral of a scalar field $f$ with respect to arc length along $C$ which can be parametrized as $g(t)$ is defined as $$\int_C f \, ds=\int_a^b f(g(t))\lVert g'(t) \rVert dt$$
@MichaelBoratko Everything is positive and the direction of the curve has nothing to do with integration with respect to arc length, so I would say the book is wrong.
OK, thanks for the help. This is the first problem on integration with respect to arc length, so I wanted to make sure I understood it correctly before doing many problems incorrectly
@MichaelBoratko They must think that the integral over the two legs of the triangle somehow cancel and the integral over the hypotenuse is in a "negative" direction. From the definitions I am seeing, that does not seem to be the case.
@skullpatrol I traversed the points in a counterclockwise manner. I believe it is a theorem that the direction does not matter in integration with respect to arc length.
@robjohn Life goes good overall. It is a few days more than a year since my first MSE post. And I've just made Deputy -- they're handing those out like candy nowadays.
@robjohn Being a moderator you could just flag a lot of stuff and then immediately declare it to be helpful. Or perhaps you don't even get a flag button to press?
@MichaelGreinecker Be logged in, and then load math.stackexchange.com/reputation -- there will be a counter of "days reached 200 rep" at the bottom of the report.
By the way, the trigger for the Mortarboard-Epic-Legendary progression is not "hitting the rep cap", but "earning 200 rep". So 19 upvotes and 1 accept will count towards Epic, but not actually hit the cap.
@MichaelGreinecker Hmm, first up, there is a definite way in which I will teach an abstract algebra course. My notes will be a geometric mean of Jacobson's text and Dummit and Foote's.
@robjohn I think I kept it cuz I thought it was an arithmetic error on the OPs part rather than an unintentional typo, but I can't really remember that post.
I think one could rewrite it as something like "How to structure a Discrete Math course" without really changing what an admissible answer would be. And if the book gets published, it will be relevant to many people.
@MichaelGreinecker But, certainly, the question here was not about teaching a course. It was about what we would like to see in their notes. Right? Moreover, it does not say who the intended audience are.
@MichaelGreinecker Well, I'll cast my vote the last... It looks answerable , perhaps.
The kernel is already a vector space, so there is no need to put "span" in front of it (though it did cause me to wonder what the "spanker" of a matrix might be...). — Gerry MyersonJul 12 at 12:35
