Analiza matematyczna funkcji wielu zmiennych
How do you find the surface area of a parametric surface? This will lead to the more general idea of a surface integral.
What we are building to
- is some surface in three-dimensional space.
- is a vector-valued function parameterizing .
- is the region of the -plane (also known as the parameter space) that corresponds with .
- is some surface in three-dimensional space.
- The surface area of can be computed with the following double integral:
- These integrals can be very labor intensive to compute.
From geometry, you might be familiar with the surface areas of a few specific shapes. For example, the surface area of a sphere with radius is .
But what if someone gives you an arbitrary surface, defined using some parametric function that maps a region of two-dimensional parameter space into three-dimensional space? How do you find its surface area?
The answer is to use a certain integral, or rather a certain double integral, which you are about to learn. This is analogous to how you can find the arc length of an arbitrary curve using a certain single integral, or the volume of a strangely shaped solid using the appropriate triple integral.
Example: Breaking down surface area
Define a parametric surface with the following function:
Let's name this surface .
Of course, with parametric surfaces, it is not enough just to specify the function which parameterizes it. We also need to know the region of the parameter space that gets mapped onto the surface. "Parameter space" is a fancy word for where the point lives, also known as the "domain". In this case, let's say it is the rectangle defined by
Let's call this rectangle . Here is what it looks like for to transform the rectangle in the parameter space into the surface in three-dimensional space.
Our strategy for computing this surface area involves three broad steps:
- Krok 1: Chop up the surface into little pieces.
- Krok 2: Compute the area of each piece.
- Krok 3: Add up these areas.
After studying line integrals, double integrals and triple integrals, you may recognize this idea of chopping something up and adding all its pieces as a more general pattern in how integration can be used to solve problems. As with those examples, our final computation will not actually involve chopping up the surface into a specific number of pieces and adding them up; we let an integral take care of that for us.
Step 1: Chopping up the surface
To start, think of chopping up the rectangle in the parameter space into many tiny little rectangles. In the drawing, I'll only chop it into a few rectangles so that we can see and reference each one, but in principle you should think of very many, really small rectangles.
For one of these tiny rectangles, you can think its width as being , a tiny change to the parameter . Similarly, think of its height as being , a tiny change to the parameter .
Now consider how the function maps one of these tiny rectangles onto the surface . In the following animation, I'll make most of the surface a faded grey, and leave just one of the tiny rectangles colored as we watch transform into .
Strictly speaking, the rectangle will become slightly curved as it is pasted onto . However, as you consider smaller and smaller rectangles, that curvature becomes more and more negligible, and we can basically treat this tiny piece as if it was flat.
In fact, as we consider smaller and smaller rectangles in the parameter space, the portions of the surface that these rectangles map to will look more and more like parallelograms.
Our first task, then, will be to find a formula giving the area of these parallelograms.
Step 2: Seeking the area of a parallelogram piece
For one of these tiny rectangles that we chopped into, let represent its lower left corner, and represent its lower right corner.
Now consider the vector pointing from to on the surface. Let's name that vector .
Concept check: If we describe the distance between and as being , which of the following expressions represents a good approximation of ?
Concept check: Take the same setup as the previous problem, but let be the upper left corner of the tiny rectangle. Let's give the vector pointing from to a name, .
If we describe the distance between and as being , which of the following best approximates ?
Okay, here's where we are so far: We are thinking about a tiny rectangle in the parameter space with the following properties
- Bottom left corner:
When you apply the function to this rectangle, you end up with what is basically a parallelogram on the surface . Based on the previous two questions, the sides of this parallelogram are determined by the vectors
Concept check: If the side lengths of a parallelogram in three-dimensional space are described with the vectors and , as pictured to the right, which of the following represents the area of that parallelogram?
Concept check: Putting all this together, when maps the little -by- rectangle with lower-left corner onto some parallelagram on the surafce , what is the area of that parallelogram?
Where this gets labor intensive
Boy is this a complicated expression. It involves two partial derivatives of a vector-valued function, taking their cross product, then taking the magnitude. It's as if someone was trying to create the most complicated expression they could imagine.
Right now we have a purely theoretical expression for the area of one of these little parallelograms:
However, if you want to get a feel for what this actually entails, I encourage you to work through it.
Work it out: Given the definition of that we started with,
evaluate the expression found in the previous question to get a function in terms of , , and .
Area of parallelogram:
Step 3: Integrating everything together
Here's where we are so far. After breaking up the rectangle of the parameter space into many tiny little rectangles, I told you that those rectangles get turned into parallelograms on the surface . Well, more accurately, they each get turned into some slightly curved piece of , which can be well-approximated by a parallelogram. The smaller your initial rectangle, the more accurate the approximation.
Then, through many computations, you found an expression for the area of one of these parallelograms:
- describes the position of the initial little rectangle.
- is its width.
- is its height.
To add up the areas of all these little parallelograms, we take a double integral of this quantity over the region . As a reminder, was defined as the region where
Using those bounds, here is the double integral representing the surface area of :
Working this out by hand seems tricky, given that finding the antiderivative of will be difficult. But using a calculator (or Wolfram Alpha), we can find the answer:
The important thing to remember here is how to construct the appropriate double integral, and to think about adding up many tiny pieces of area on the surface itself.
Summary: This ain't easy
Generalizing everything we did in the previous example, the surface area of our parametric surface is expressed using the integral
where is described using a parametric function applied to a region of the -plane.
You've already had a glimpse of this, but it's worth pointing out that this can be a really complicated thing to compute.
- First you have to take two partial derivatives of vector-valued functions, which if you count each component includes partial derivatives in total.
- You then have to take the cross product of these two partial derivative vectors, which itself requires taking a determinant whose components are vectors and functions.
- Then you have to compute the norm of that cross product.
- After all that, there is still a double integral ahead of you. And remember, just setting up a double integral isn't always easy, especially if the region you are integrating over is not rectangular.
- And all this is assuming you already know the function and the region . Sometimes you are just given a surface which is implicitly defined, like a sphere defined by . In that case you need to find a function which parameterizes this surface, as well as which specific region of the parameter space corresponds to the surface.
The key when going through all of this is to stay organized, and be patient. One way to think about it is that setting up and computing just one of these surface area integrals is akin to doing practice problems in single-variable calculus.
The thought process that goes into all of this is actually very useful for thinking about surfaces and three-dimensional geometry in general, beyond the specific case of computing surface area. For example, how do you think computer graphics works? Quite often, displaying a three-dimensional figure involves subdividing a surface into polygons, and getting the computer to display those polygons. Even if this never involves performing a surface area integral, per se, the reasoning associated with how to do this is remarkably similar, using cross products of partial derivatives, etc.
If you want to practice this more, the next article walks through another full example. If you do choose to work through it, prepare to mark up a lot of paper.