# graph of function of 2 variables

by jahlin
Tags: function, graph, variables
 P: 21 if a function is definied by z=f(x,y)..the graph of the function is a surface without thickness right ?i cant really differentiate between a 3d and 2d graph..both look like surfaces.
 P: 74 Provided all the numbers are real, yes. Every point on the X-Y plane is mapped to the "surface" by a translation along the Z-axis. The value of the translation depends on the result of f(x, y). Similarly in 2-D, every point on the X-axis is mapped to a "curve" by a translation in the Y-direction. The value of the translation depends on f(x).
 P: 21 the graph of function of 3 variables f(x,y,z) is a surface with thickness?
P: 74

## graph of function of 2 variables

No. f(x,y,z) is a more general case of z=f(x,y).

Consider 2 dimensions first. y=f(x) gives a curve with a few interesting properties, one of which is called one-to-one correspondence. That means that you can draw a vertical line anywhere on your graph, and it will cross the curve exactly once. (The same need not be true for a horizontal line.)

If I relax that restriction and re-arrange the equation, I can write f(x,y)=const (still in 2D here). This still gives me a curve, but I don't have the one-to-one correspondence anymore.

For example, x^2 + y^2 = 1 is the equation for a unit circle. That is, all the (x,y) pairs which satisfy that equation lie on a circle centered at origin with unity radius.

Now extend the thinking to 3D. Unsurprisingly, x^2 + y^2 + z^2 = 1 is the equation for a sphere centered at origin and with unity radius. This is an equation in 3 variables of the form f(x, y, z) = const. If we were to rewrite that equation in the form z = f(x, y), we would see that f(x, y) = sqrt(x^2 + y^2). This equation has two solutions; one for the "top half" of the sphere, and one for the "bottom half". That is because the square root function is ambiguous: say y=sqrt(x), then x^2=y, but also (-x)^2=y.

So it is the same difference in 2D and 3D. You can draw a line anywhere, so long as it's parallel to the z-axis and it will intersect the surface exactly once in the case of z=f(x,y). No so if f(x,y,z)=const. We relax the correspondence rule and can get all sorts of wild shapes.
 P: 21 as you said every point on the X-Y plane is mapped to the "surface" by a translation along the Z-axis. what abt the graph of 3d every point is mapped to a surface by a translation along what axis?
 P: 74 That is in 3D. In 2D, an x-value (a magnitude, strictly speaking) is mapped onto a curve by a translation along the Y-axis. In 3D, an x-y pair (a point) is mapped onto a surface by a translation along the Z-axis. There is no notion of a surface in 2D, only points and curves. A surface is strictly a 3D concept.
Math
Emeritus