# what's a surface?

• Jul 31st 2014, 04:50 PM
catenary
what's a surface?
As I understand a surface is any function in three dimensions, or a function that contains 3 variables (e.g. f(x,y,z) = x + y + z). However my math teacher gave this example:

Quote:

Find the angle between the surfaces:
$A = x^2 + y^2 + z^2 = 9$
$B = z = x^2 + y^2 - 3$
at the point P(2,-1,2)

gradient of A = 2xi + 2yj +2zk
gradient of B = 2xi + 2yj - k

at the point P:
gradient of A = 4i - 2j + 4k
gradient of B = 4i - 2j - k

gradA dotted with gradB = $|{gradA}| |{gradB}| cos(theta)$
then theta = 54.4 degrees
I am wondering if the given( $A = x^2 + y^2 + z^2 = 9$ and $B = z = x^2 + y^2 - 3$), really can be considered surfaces even though they are equations, and if the example is correct...
• Jul 31st 2014, 09:09 PM
Deveno
Re: what's a surface?
A surface is a 2-dimensional manifold. Unfortunately, that sort of begs the question. There are different ways to represent surfaces, which may give misleading impressions about their dimensionality. Some different ways surfaces may be obtained:

Explicitly, as the GRAPH of a function $f: \Bbb R^2 \to \Bbb R$, typically written $z = f(x,y)$. For example:

$z = x^2 + y^2$, which is an elliptic (circular, actually) paraboloid.

Parametrically, as the IMAGE of a (non-degenerate) function $p: \Bbb R^2 \to \Bbb R^n$. For example the sphere can be described as the image of:

$p(u,v) = (r\sin u\cos v,r \sin u \sin v, r\cos u)$

Implicitly, as the set of zeros for a function $g: \Bbb R^3 \to \Bbb R$. An example of this is another version of the sphere:

$g(x,y,z) = x^2 + y^2 + z^2 - r^2$, when $g(x,y,z) = 0$, we obtain the (implicitly defined) functions:

$f_1(x,y) = \sqrt{r^2 - x^2 - y^2}$

$f_2(x,y) = -\sqrt{r^2 - x^2 - y^2}$

which are "implied" by the condition: $x^2 + y^2 + z^2 = r^2$.

In your problem, the latter situation is the case for $A$, the "$g$" is: $g(x,y,z) = x^2 + y^2 + z^2 - 9$., and the first situation is what we have for $B$, where $f(x,y) = x^2 + y^2 - 3$.

It is not uncommon for different ways to represent surfaces to be used in a "mix and match" scenario, depending on what facilitates calculation.
• Aug 1st 2014, 04:11 AM
catenary
Re: what's a surface?
Thank you so much Deveno, that was very enlightening. So if a surface has been represented in a certain way then can it be converted to any of the other forms of representing it? So if I converted B in my example from: $B = z = x^2 + y^2 - 3$ to $B = x^2 + y^2 - z = 3$
then B becomes implicitly defined where the g is $g(x,y,z) = x^2 + y^2 -z -3$?
Also, when you say: $f:R^2---->R$, does this translate as: "the domain of the set of real numbers in two dimensions mapped as a set of real numbers in one dimension"?
$p:R^2---->→R^n$, "the domain of the set of real numbers in two dimensions mapped as a set of real numbers in n dimensions"?
$g:R^3---->→R$, "the domain of the set of real numbers in three dimensions mapped as a set of real numbers in one dimension"?
• Aug 1st 2014, 04:33 AM
HallsofIvy
Re: what's a surface?
Quote:

Originally Posted by catenary
As I understand a surface is any function in three dimensions, or a function that contains 3 variables (e.g. f(x,y,z) = x + y + z).

This is wrong. Of course, to be completely correct, a "surface" is a geometric object, not a function or equation at all, but you mean the surface "can be represented by" in a given coordinate system. A two dimensional surface, in a three dimensional coordinate system cannot be represented by "f(x, y, z)" because you would need a fourth dimension to represent the value of f itself. What you want is an equation of the form z= f(x,y) or, more generally, f(x, y, z)= constant.

Quote:

However my math teacher gave this example:

I am wondering if the given( $A = x^2 + y^2 + z^2 = 9$ and $B = z = x^2 + y^2 - 3$), really can be considered surfaces even though they are equations, and if the example is correct...
Yes, a surface requires an equation not just a function. Your "understanding" was incorrect.