Table of contents 
Introduction 
Graph of a Plane 
Domains of Functions of more than one variable 
Level Curves or Contour Curves 
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Let's study some of the basic ideas about the functions of more than one variable.
First, remember that graphs of functions of two variables, z = f (x,y) are surfaces in threedimensional space. For example here is the graph of z= 2x^{2} + 2y^{2}  4
This is an elliptic paraboloid and is an example of a quadric surface. We saw several of these in the previous section.
Another common graph that we’ll be seeing quite a bit in this course is the graph of a plane. There is a convention for graphing planes that will make them a little easier to graph and hopefully visualize.
For example: Let’s graph the plane given by, f(x,y) = 12  3x  4y.
Now, to extend this out, graphs of functions of the form w = f (x,y,z) would be fourdimensional surfaces. Of course, we can’t graph them, but it doesn’t hurt to point this out.
Next, let's discuss the domains of functions of more than one variable.
Recall that domains of functions of a single variable, y = f(x), consisted of all the values of x that we could plug into the function and get back a real number. Now, if we think about it, this means that the domain of a function of a single variable is an interval (or intervals) of values from the number line or onedimensional space.
The domain of functions of two variables, z = f (x,y), are regions from twodimensional space and consist of all the coordinate pairs, (x,y), that we could plug into the function and get back a real number.
Example 1  Determine the domain of each of the following.
Ans.
(a) In this case we know that we can’t take the square root of a negative number so this means that we must require, x+y > 0.
Here is a sketch of the graph of this region.
(b) This function is different from the function in the previous part. Here we must require that,
x > 0 and y > 0 and they really do need to be separate inequalities. There is one for each square root in the function. Here is the sketch of this region.
(c) In this final part we know that we can’t take the logarithm of a negative number or zero. Therefore we need to require that, and upon rearranging we see that we need to stay interior to an ellipse for this function. Here is a sketch of this region.
Note that domains of functions of three variables, w = f(x,y,z), will be regions in threedimensional space.
Example 2  Determine the domain of the following function,
Ans.
In this case, we have to deal with the square root and division by zero issues. These will require,
x^{2}+y^{2}+16 > 0 ⇒ x^{2} + y^{2} + z^{2} > 16
So, the domain for this function is the set of points that lies completely outside a sphere of radius 4 centred at the origin.
Example 3  Find the domain of each of the following functions.
f(x,y)=√1−x2−y2.
g(x,y)= 1/√1−x2−y2 .
h(x,y)= 1/ yx^{2}
Ans.
(1) The domain of f is the set of points (x,y) such that 1−x^{2}−y^{2}≥0. We recognize x^{2}+y^{2}=1 as the equation of a circle of radius 1 centred at the origin, and so the domain of f consists of all points which lie on or inside this circle (see below).
(2) We now require that 1−x^{2}−y^{2}>0. The domain of g therefore contains all points (x,y) such that x^{2}+y^{2}<1; that is, all points which lie strictly inside the unit circle (see below).
(3) We see that h(x,y) is undefined for x=y^{2}. The domain of h therefore consists of all points in the xyplane except those which satisfy y=±√x (see below).
The next topic that we should look at is that of level curves or contour curves.
The level curves of the function z = f(x,y) are twodimensional curves we get by setting z = k, where k is any number. So the equations of the level curves are f(x,y) = k.
Let’s do a quick example of this.
Example 4  Identify the level curves of Sketch a few of them.
Ans.
Let’s take a quick look at an example of traces.
Example 5 Sketch the traces of f (x,y) = 10  4x^{2}  y^{2} for the plane x = 1 and y = 2.
Ans.
We’ll start with x = 1. We can get an equation for the trace by plugging x = 1 into the equation. Doing this gives,
and this will be graphed in the plane given by x= 1.
Below are two graphs. The graph on the left is a graph showing the intersection of the surface and the plane given by x = 1. On the right is a graph of the surface and the trace that we are after in this part.
For y =2 we will do pretty much the same thing that we did with the first part. Here is the equation of the trace,
and here are the sketches for this case.
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