Functions of Multiple Variables

1. FUNCTION
 

The most common mental mode of a function is a machine. When you put some input in to the machine, you will always get the same output. Most first year calculus dealt with the function where the input was a single real number and the output was a single real number too. The study of advanced calculus begins by modifying this idea. I illustrate this idea with an example below:

f (x, y) = x² + y²

^{For each value of x and y there is one value of  f (x, y). Try it with nominal x= 2 and y= 3.}

^{Unfortunately, plugging in random points doesn't give much enlightenment as to the behavior of a function. Perhaps a more visual model would help.}

^{2. THREE DIMENSION}

^{In the previous section we saw that plugging random points in to function of two variables gave almost no enlightening information about the function itself. A far superior way to get a handle on a particular function is to picture its graph.}

^{Coordinate systems will play a crucial role. To plot a point with two coordinates such as (x,y) = (2,3) the first step is to draw two perpendicular axes and label them x and y. Then locate a point 2 units from the origin on the x-axis and draw a vertical line. Next, locate a point 3 units from the origin on the y-axis and draw a horizontal line. Finally, the point (2,3) is at the intersection of the two lines you have drawn.}

^{To plot a point with three coordinates, the steps are just a bit more complex. Plot the point (x, y, z) = (2, 3, 2). First, draw three mutually perpendicular axes. Then you immediately notice that this is impossible to do on a sheet paper. The best you can do is two perpendicular axes, and a third at some angle to the other two. With practice you will start to see this third axis as a perspective rendition of a line coming out of the page. When viewed, this way it will seem like it is perpendicular.}

^{The way to remember it is by the right hand rule. What you want is to be able to position your right hand so that your thumb is pointing along the z-axis and your other fingers sweep from the x-axis to the y-axis when you make a fist. If the axes are labeled consistent with this then we say you are using a right handed coordinate system.}