Would someone be able to walk me through what this definition means? line integrals

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Homework Statement



For some scalar field f : U ⊆ Rn → R, the line integral along a piecewise smooth curve C ⊂ U is defined as

[tex]\int_C f\, ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)|\, dt[/tex]

where r: [a, b] → C is an arbitrary bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C and a<b .
 
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  • #2
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sorry, for breaking the format. I will put my attempt in the next post. Just wanted to have the stuff rendered on the screen!
 
  • #3
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so the first part is confusing me quite a bit. What does [itex]\mathbf{R}^n\rightarrow \mathbf{R}[/itex] mean?

when I hear "scalar field" I would think of it as a field of some scalar number in space. Maybe like potential? I know that potential energy is a scalar, so I guess potential (i'm thinking electric potential) would be a scalar as well. So at each point in space there could be a different potential. Is this a "scalar field"?

so anyways I think I get that the piecewise smooth curve C is a subset of U. that just means that the curve is somewhere in the potential field right?
 
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  • #4
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basically I'm not asking for someone to explain all of line integrals to me. I have been evaluating them for awhile and have kind of a basic idea of what they are. I'm looking for what this notation is "saying". I am getting lost in all the symbols!
 
  • #5


##f:U\subseteq\mathbb R^n\to\mathbb R## means that ##f## maps a subset ##U## of the n-dimensional real space ##\mathbb R^n## to the set ##\mathbb R## of reals. So each vector ##\mathbf v\in U## is mapped onto a scalar ##f(\mathbf v)\in\mathbb R##.

Your "potential" example is a good one, and the "field" in this sense is the mapping from ##U## to ##\mathbb R##, and not the set ##U## itself, assigning a "potential" to each point in ##U##. And the curve ##C## lies somewhere in ##U##. It's independent of the field ##f##.
 
  • #6


Second part: if you integrate a function ##f:\mathbb R\to\mathbb R##, say from ##a## to ##b##, you're forming a kind of sum,
$$
\int_a^bf(x)\,\mathrm dx = \lim_{\delta\to0}\delta\cdot\left(f(a)+f(x+\delta)+f(x+2\delta)+ \ldots +f(b)\right).
$$
There's only one path from ##a## to ##b##, and it leads through each intermediate point ##a+n\delta##. If you want to integrate from one point in ##\mathbb R^n## to another, you have to specify which path you are going to take. That's what ##C## describes.
 
  • #7
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ok hold on a second. I am still a little bit confused about your first post (and thanks a ton for making it!)

I'm struggling to understand what you mean when you say each vector that is an element of U is mapped onto a scalar field.
 
  • #8
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so like going with that first analogy. The electric field is a vector field right? each point in an electric field also has a potential. so is the electric field somehow mapped onto the potential field?
 
  • #9


Each vector v in U has some potential, which we call f(v). The "field" here is the function f. In mathematics, a function is also called a "mapping", and each vector v is "mapped" onto the scalar f(v).

Your second example, an electric field ("field" in the physical sense) can be seen as a (mathematical) vector field, i.e. a mapping g that assigns a vector g(v) to each point v in space. This vector g(v) corresponds to the electromagnetic force that would act on a particle at position v.

These two fields, the scalar field f and the vector field g, are related, but not the same. And you can't "map" one to the other, since both are mappings themselves.
 
  • #10
Ray Vickson
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so like going with that first analogy. The electric field is a vector field right? each point in an electric field also has a potential. so is the electric field somehow mapped onto the potential field?
Potential is a perfect example: for each point (x,y,z) in R^3 you get a number V(x,y,z). So V:R^3 -> R. This notation just means that for each point p = (p_1, p_2, ..., p_n) in U you get a number f(p) = f(p_1,p_2,...,p_n).
 

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