eli5 what are line integrals?


I know a lot about them, formulae and all, but, I wanted to know if what I am thinking is right or not.

So integral of some function f(x) would give us the area under the curve of that function. We can use regular integral in this case

Line integral is basically like normal integral but for 2 dimensional curves and it too gives us the area under that curve (height of that curve at x *dx integrated). Am I right so far?

Also, previously I imagined that for line integral the functions would be like f(x,y), what f there is a function which has more than two variables? Would line integral hold good for them too or is line integral just for two variable functions.

In: 2

Finally made it! I could not post the question in all bold for some reason! weird






Line integrals give a sum of some quantity through a vector field along a certain path. Let’s say you have an EM field and you wanna know how much work it takes to move a test particle in a given through that force field, I guess you could say is a good intuition.

That’s probably the most common use I can think of in physics really.

There is an animation on the wikipedia article that I think does a really good job of showing how a line integral works. See the link below.


To answer some of your questions (but it will be hard to be truly eli5), a line integral is an integral of a scalar field, or a vector field defined on some domain. The domain is not limited to be 2 dimensions. That is, it is well defined and no problem to take a line integral of a field defined on a three dimensional domain for instance. (An example of this would be taking the integral of the electric field around a closed loop. This integral gives you the rate of change of magnetic flux through the surface that the loop bounds, from Maxwell’s equations).

What makes it a line integral, is not whether it is a two dimensional domain or not, but that the integral is taken over a one dimensional path (that is, there is a path that depends, or is parameterized, by one variable, like time or arc length).*

The value of the scalar field or vector field at each point along the path then contributes to the final value of the integral in the same way as normal calculus (ie. limit of the sum of many small portions of the field at each position of the path).

*For more detail on that point, you might imagine a particle that moves in some 3D volume. The path that particle takes can be described by it’s x, y and z coordinates, each of which is a function of time. So the position of the particle is fully determined at a given time, and can traced over the full length of time. The line integral can be taken over that path, and you have a 1 dimensional parameterized path, say r(t), where r is the position vector, and the field has a particular value at each point of the path (a simple number if a scalar field, or a vector if a vector field).