graph of function of 2 variables

jahlin

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.

p1ayaone1

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).

jahlin

the graph of function of 3 variables f(x,y,z) is a surface with thickness?

p1ayaone1

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.

jahlin

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?

p1ayaone1

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.

HallsofIvy

What do you mean by "3d graph" and "2d graph"? Are you refering to the dimension of the object itself or the dimension of the coordinate system it is graphed in? A surface is a two dimensional object that (unless it happens to be a plane) must be graphed in three dimensions.

Since z= f(x,y) has two independent variables and one dependent, its graph is a two dimensional surface graphed in a three dimensional coordinate system.

jahlin

I was confusing the dimensions of an object with the graph. I understood it now. Thanks HallsofIvy and playaone1 for clarifying.