What are line and surface integrals?

[cdotter]

You can say an integral is the area under a curve and the derivative is the slope. What are the equivalents for line and surface integrals? I've tried google and wikipedia but I can't find a dumbed down version. I know that line integrals are related in some way to arc length since a line integral is $\int_{S}f(x,y) \, ds$ where ds is the derivative of arc length, but I don't understand how it really ties together.

 

[lawsofform]

Remember that the integrals you are probably used to seeing are of one-dimensional functions, so thinking of it as an area is somewhat misleading. Instead, think of it as "adding up" the values at every point of a function.

Now for a line integral, instead of adding up the value of a function from point A to point B, you take a some line (or if you like, path or curve) on a function represented by a surface, and add up all the values of the function along that line. It doesn't really have very much to do with arc length, except you might have to compute arc length while computing a line integral.

The surface integral is the same sort of thing, except that instead of integrating along a line, you are integrating over an area.

 

[cdotter]

Thanks, that made it much clearer.

 

[tiny-tim]

hi cdotter!

another way of looking at it:

an integral over one variable is the area under a curve

an integral over two variables is the volume under a surface

for a line integral, imagine that at each point of the line you're drawing a perpendicular line equal to the value there … that gives you a graph of the value, and the line integral is the area under the graph

similarly, for a surface integral, imagine that at each point of the surface (it can be curved) you're drawing a perpendicular line equal to the value there … that gives you a graph of the value, and the surface integral is the volume under the graph

(and a volume integral would be the mass of a an object with that density occupying that volume)

 

[cdotter]

hi cdotter!

another way of looking at it:

an integral over one variable is the area under a curve

an integral over two variables is the volume under a surface

for a line integral, imagine that at each point of the line you're drawing a perpendicular line equal to the value there … that gives you a graph of the value, and the line integral is the area under the graph

similarly, for a surface integral, imagine that at each point of the surface (it can be curved) you're drawing a perpendicular line equal to the value there … that gives you a graph of the value, and the surface integral is the volume under the graph

(and a volume integral would be the mass of a an object with that density occupying that volume)

I don't understand your explanation with perpendicular lines?

 

[LCKurtz]

I don't understand your explanation with perpendicular lines?

That isn't my favorite interpretation either, so don't feel bad. I like to give physical applications to motivate line integrals. For example, suppose you have a curve C in 2 or 3 dimensions. It might represent the location of a wire with linear mass density d(x,y) or d(x,y,z). Then the line integral

$$\int_C d(x,y) ds$$

represents the mass of the wire. If d(x,y) = 1 you get the length. Or if the curve is in a force field F, the line integral

$$\int_C \vec F \cdot d\vec R$$

represents the work done by the force if an object is moved along the curve.

Similarly surface integrals my represent area or mass of a surface or they may represent "flux" flow through a surface such as in heat transfer.

 

[cdotter]

That isn't my favorite interpretation either, so don't feel bad. I like to give physical applications to motivate line integrals. For example, suppose you have a curve C in 2 or 3 dimensions. It might represent the location of a wire with linear mass density d(x,y) or d(x,y,z). Then the line integral

$$\int_C d(x,y) ds$$

represents the mass of the wire. If d(x,y) = 1 you get the length. Or if the curve is in a force field F, the line integral

$$\int_C \vec F \cdot d\vec R$$

represents the work done by the force if an object is moved along the curve.

Similarly surface integrals my represent area or mass of a surface or they may represent "flux" flow through a surface such as in heat transfer.

Couldn't you just use arc length if it's a linear mass density??

 

[LCKurtz]

Couldn't you just use arc length if it's a linear mass density??

No. The point is that the density may vary along the wire and the integral gives its mass.

Also, what I meant by "linear mass density" is that density is given in units of mass/length. A wire might have density of 5 grams/meter at one end building up to 10 g/m at the other. Or for a surface, you might give density in units of mass/area. As in a sheet of aluminum with density 2kg/square meter. (I just made the numbers up as examples).

 

[cdotter]

No. The point is that the density may vary along the wire and the integral gives its mass.

Also, what I meant by "linear mass density" is that density is given in units of mass/length. A wire might have density of 5 grams/meter at one end building up to 10 g/m at the other. Or for a surface, you might give density in units of mass/area. As in a sheet of aluminum with density 2kg/square meter. (I just made the numbers up as examples).

So in other words, the line integral is just a summation of all the linear mass densities along the curve or wire * the length of the curve or wire, which equals the mass of the wire?

 

[LCKurtz]

So in other words, the line integral is just a summation of all the linear mass densities along the curve or wire * the length of the curve or wire, which equals the mass of the wire?

If you mean the incremental lengths, yes. The approximating sum would look like

$$\sum_{i=1}^n \delta(x_i^*,y_i^*)\Delta s_i$$

the sum of densities times incremental lengths, which approximates the total mass.

 

[7thSon]

cdotter, I assume you are just starting out with vector calculus. I think for the time being it may help to let go to some extent of the notion that there is some physical meaning of the surface integral or a line integral of a random function.

If I look at a random function defined in a field, and take a line integral of that function along some path, you are physically doing the same thing with any other Riemann sum limit... you are adding up bits and pieces of that function along the line. There is only a physical interpretation depending on the problem you are considering.

For example, if you do a line integral along some path in a conservative force field, you can calculate the potential difference or work done on an object. In general, a line integral of some random function f(x,y,z) along some path in R3 might not have a physical interpretation.

 

[cdotter]

cdotter, I assume you are just starting out with vector calculus. I think for the time being it may help to let go to some extent of the notion that there is some physical meaning of the surface integral or a line integral of a random function.

If I look at a random function defined in a field, and take a line integral of that function along some path, you are physically doing the same thing with any other Riemann sum limit... you are adding up bits and pieces of that function along the line. There is only a physical interpretation depending on the problem you are considering.

For example, if you do a line integral along some path in a conservative force field, you can calculate the potential difference or work done on an object. In general, a line integral of some random function f(x,y,z) along some path in R3 might not have a physical interpretation.

I think so too because it just gets more confusing as I try to understand it.