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 2 Find the Fourier series for on .
Because of the piece-wise nature of the function the work for the coefficients is going to be a little unpleasant but let’s get on with it.
At this point it will probably be easier to do each of these individually.
So, if we put all of this together we have,
So, we’ve gotten the coefficients for the cosines taken care of and now we need to take care of the coefficients for the sines.
As with the coefficients for the cosines will probably be easier to do each of these individually.
So, if we put all of this together we have,
So, after all that work the Fourier series is,
At this point it will probably be easier to do each of these individually.
So, if we put all of this together we have,
So, we’ve gotten the coefficients for the cosines taken care of and now we need to take care of the coefficients for the sines.
As with the coefficients for the cosines will probably be easier to do each of these individually.
So, if we put all of this together we have,
So, after all that work the Fourier series is,
As we saw in the previous example there is often quite a bit of work involved in computing the integrals involved here.
The next couple of examples are here so we can make a nice observation about some Fourier series and their relation to Fourier sine/cosine series.
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