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What are some geometric meanings behind line integrals? I know if you have a curve on the xy plane and you are given a function $f(x,y)$ then the geometric meaning is a "curtain drawn" from the function (surface) to the curve below. This is a planer curve.

However what about when we just have a helix? Similarly what the geometric meaning (or explanation) for the line integral involving a vector field $F$ and a curve. $$\int_c{F\cdot ds}$$

Here is an example:

Let $c(t)= \sin{t}, \cos{t}, t$ from 0 to $2\pi$. let the vector field $F$ be defined by $$F(x,y,z)=x\hat{i} + y\hat{j} + z\hat{k}$$ Compute $\int_c{F\cdot ds}$

Any links to pdfs and other resources helping me understanding would be very helpful!


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2 Answers 2

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There are several really good resources. But for all of your calculus needs, Paul's Online Notes is typically very well written. The link for his beginning of an explanation of line integrals is

Also, Paul does a very good job at explaining many other facets of calculus. Enjoy.

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I actually have his entire calculus and differential equations printed out. However his stuff is good for basic understanding but I have a pretty tough midterm coming up so I am looking for more rigorous explanations. I know that kind of contradicts my question. – Tyler Hilton Nov 13 '11 at 1:44

When a line integral involves a vector field, it actually doesn't have a fixed geometric meaning. For example, by the Green's theorem, you can use a line integral calculated over a boundary curve to calculate the double integral taken over the area (region) bounded by that curve, which can be interpreted geometrically as the volume of the solid defined by the $(x,y)$-plane and that region (a function $f(x,y)$). Here a line integral (of a vector field) is being used to calculate volume, albeit indirectly, while in the example you stated, the line integral (of a scalar field) is used to calculate the 'area' of the curtain. So the line integral of a vector field may not actually have a geometric meaning. However, a meaning can be assigned to it according the situation it is being used in. One such example is that

$$\int_C F⋅ds$$

is equivalent to work done when a vector force $F$ moves an object along a path defined by the curve $C$, and work done can be thought of as the area under a force-displacement graph (note: force-displacement graph, not force-distance, while $ds$ is a distance).

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