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I learned very recently about line integrals in my class, and I've been given the definition: $$\int _\Gamma \vec {F} \cdot \vec {ds}=\int _a^b \vec F(\vec \alpha(t))||\alpha'(t)||dt$$

Where $\alpha$ is an injective parametrization of $\Gamma$.

My questions are:

  1. Given this definition, it seems like the only way to calculate a line integral (besides easy ones where you may use geometric tricks) is to parametrize the curve of domain, is that so?
  2. What kind of information does the line integral give about a vector field ($\vec F: \Bbb R^3\to \Bbb R^3$)?
  3. Why do we need a new symbol for "closed path" line integrals?

I know that when talking about functions $g:\Bbb R^2 \to \Bbb R$, it gives the area between the curve and the function, but couldn't get the intuition on what it tells for functions from $\Bbb R^n \to \Bbb R^m$.

Also, if we were treating functions with units (as in physics) what units would the line integrated function get?

YoTengoUnLCD
  • 13,384

1 Answers1

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  1. Yes, but sometimes people don't realize that's what they're doing.
  2. It essentially tells you how much your oriented path is going in the same direction as the ambient vector field. If you're familiar with mechanical work, think of work along some curvy path in a gravitational field.
  3. It's sometimes useful to know that the curve is closed, but you don't have to use the notation if you don't want.

units of $\int F \cdot ds = $ units of $F \cdot s$. So if $F$ is in Newtons and $s$ is meters, then you'd get Newton-meters (or Joules).

user268545
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