Analiza matematyczna funkcji wielu zmiennych
Learn how to compute and interpret line integrals, also known as path integrals or curve integrals.
Do czego zmierzamy
- In the same way that an ordinary integral has you walking along the axis and adding up certain tiny quantities as you go, a line integral has you walking on a curve the -plane, adding up certain quantities depending on the multivariable function .
- If a curve is parameterized by a vector value function between the values and , the line integral is written as follows:
- In this case, is a scalar valued function, so we call this process "line integration in a scalar field", to distinguish from a related idea we'll cover next: line integration in a vector field.
What is a line integral
In the last article, I introduced some compact notation for an arc length integral:
- The term represents a tiny change in length along the curve.
- is just a name we give to the curve. Putting it at the base of the integral, like this , is a way to postpone the need to place actual bounds on the integral.
Line integrals extend this idea by placing a multivariable function inside the integral,
You can think of this integral as saying
"Hey, while you're walking along the curve taking tiny steps of size , rather than just adding up the sizes of those steps, multiply each step size by the value of the function at the point where you are standing."
The following animation relates this to the more familiar idea of finding the area under a curve. Imagine a curtain draped under the three-dimensional graph of along the curve in the -plane. The line integral gives the area of that curtain. (The initial image is a colored contour plot of the function ).
You imagine the area of that curtain being broken up into infinitely many infinitely thin rectangles. The infinitesimal base of each rectangle is , the size of a tiny step along the curve. The height of each rectangle is , the height of the graph of at that point.
Vector notation for line integrals
At the end of the animation above, the line integral is written like this:
Let's break down what each part of this means.
The parameterization of
is some vector-valued function which parameterizes the curve . In two dimensions, it might look something like this:
The bounds of the integral, and , are values for which determine where the curve starts and ends.
Essentially this means that as the value of ranges between and , the tip of the vector traces over the curve .
Composing with the parameterization
Evaluating means taking the components and of , and plugging them in as the inputs of :
You can think of it this way: A given value of puts us at some point on the -plane, expressed as the tip of the vector . Then gives the value of the function at that point on the plane.
is the magnitude of the derivative (times )
The term , representing a tiny step along the curve, is expanded as , the magnitude of the derivative of times .
Intuitively, this is because the derivative answers the question "what happens if you nudge the input slightly by a value ?" The answer is that the output gets nudged along the vector in the -plane. The magnitude of this nudge in the output space gives you the size of a step along the curve, as a function of the size of the step in the parameter space .
You can also see this as the compact vector-notation way of saying :
Putting all this together, we get the vector representation of a line integral:
Example 1: Compute a simple line integral
Let be one quarter of a circle with radius centered at the origin. The quarter in the first quadrant to be specific.
We can describe this quarter circle parametrically with the following function:
If we let range from to , this traces around .
Now defined the multivariable function as
Our goal is to compute the line integral
The following video shows what the "curtain" beneath the graph of along the curve looks like. The translucent white plane is the graph of , and the blue surface is the curtain whose area we are computing.
Step 1: Write in terms of
Given that the size of a tiny step along the curve is given by the magnitude of the derivative of times ,
solve for in our example.
For reference, the parameterization of our curve is
Step 2: Write in terms of
What is in this case?
Step 3: Write the integral completely in terms of and solve
From the previous two steps, our integral becomes
Since our parameterization of has running from to , these are the bounds of the integral. Now solve the integral.
Example 2: More intricate practice
In principle, line integration is not too bad once you get the hang of it. You just have to know how to expand the term , and to rewrite the inputs of the function in terms of the parameterization. Knowing how to set this up just takes a little practice, which is what we're doing here.
That said, line integrals can be a real pain to actually compute. Most of them end up in a state where you need to plug the integral into a computer to get an answer, but even when the integral is solvable the numbers involved can quickly grow cumbersome.
The following example shows how even for relatively simple functions, computing a line integral to its completion can be a very involved problem. You will definitely want to have a pencil and scratch paper ready if you choose to go through this example.
Let be the curve defined by the function
on the interval .
Let be the function
Compute the line integral
Step 1: Write in terms of
Step 2: Replace with
What do you get when you plug the components of into ?
Step 3: Solve the integral
Plug the answers from the previous two steps into the integral, then solve. Since the curve is defined for , and our integral is now given with respect to , the bounds of the integral are and .
(Warning, this gets super ugly to compute, so feel free to put it in a calculator when the time comes.)
Notice, the hard part of this problem is not the new principles of line integration, knowing how to expand and all that. What makes it hard is that the complexity of the terms involved quickly blows up.
(Also, if you think this is bad, just wait until we get to surface integrals.)
Line integrals in a scalar field
In everything written above, the function is a scalar-valued function, meaning it outputs a number (as opposed to a vector). There is a slight variation on line integrals, where you can integrate a vector-valued function along a curve, which we will cover in the next article.
To distinguish these ideas, everything we just covered is typically called line integration in a scalar field and the alternative is referred to as line integration in a vector field. The term "scalar field" is just another way of thinking about what a multivariable function does: It associates each point in the -plane with some scalar (i.e. number), so that the entire plane is like a field of numbers just waiting for someone to wander along a path through that field and integrate those values.