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Graph of function of 2 variables 
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#1
Nov1910, 05:15 PM

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.



#2
Nov1910, 06:10 PM

P: 74

Provided all the numbers are real, yes. Every point on the XY plane is mapped to the "surface" by a translation along the Zaxis. The value of the translation depends on the result of f(x, y).
Similarly in 2D, every point on the Xaxis is mapped to a "curve" by a translation in the Ydirection. The value of the translation depends on f(x). 


#3
Nov1910, 06:33 PM

P: 21

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



#4
Nov1910, 07:02 PM

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 onetoone 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 rearrange 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 onetoone 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 zaxis 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. 


#5
Nov1910, 07:16 PM

P: 21

as you said every point on the XY plane is mapped to the "surface" by a translation along the Zaxis.
what abt the graph of 3d every point is mapped to a surface by a translation along what axis? 


#6
Nov1910, 07:37 PM

P: 74

That is in 3D.
In 2D, an xvalue (a magnitude, strictly speaking) is mapped onto a curve by a translation along the Yaxis. In 3D, an xy pair (a point) is mapped onto a surface by a translation along the Zaxis. There is no notion of a surface in 2D, only points and curves. A surface is strictly a 3D concept. 


#7
Nov2010, 07:08 AM

Math
Emeritus
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Thanks
PF Gold
P: 39,545

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. 


#8
Nov2110, 11:16 AM

P: 21

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



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