In this section we need to take a look at the equation of a line in R 3 R 3 . As we saw in the previous section the equation y = m x + b y = m x + b does not describe a line in R 3 R 3 , instead it describes a plane. This doesn’t mean however that we can’t write down an equation for a line in 3-D space. We’re just going to need a new way of writing down the equation of a curve. So, before we get into the equations of lines we first need to briefly look at vector functions. We’re going to take a more in depth look at vector functions later. At this point all that we need to worry about is notational issues and how they can be used to give the equation of a curve. The best way to get an idea of what a vector function is and what its graph looks like is to look at an example. So, consider the following vector function. → r ( t ) = ⟨ t , 1 ⟩ r → ( t ) = ⟨ t , 1 ⟩ A vector function is a function that takes one or more variables, one in this case, and returns a vector. Note as we
Example 4 Find the Fourier cosine series for on .
The integral for is simple enough but the integral for the rest will be fairly messy as it will require three integration by parts. We’ll leave most of the details of the actual integration to you to verify. Here’s the work,
The Fourier cosine series for this function is then,
The Fourier cosine series for this function is then,
Finally, let’s take a quick look at a piecewise function.
Example 5 Find the Fourier cosine series for on .
We’ll need to split up the integrals for each of the coefficients here. Here are the coefficients.
For the rest of the coefficients here is the integral we’ll need to do.
To make life a little easier let’s do each of these separately.
Putting these together gives,
So, after all that work the Fourier cosine series is then,
For the rest of the coefficients here is the integral we’ll need to do.
To make life a little easier let’s do each of these separately.
Putting these together gives,
So, after all that work the Fourier cosine series is then,
Note that much as we saw with the Fourier sine series many of the coefficients will be quite messy to deal with.
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