Surface integration and parametrization confusion Suppose you have a surface S  in $R^3 $. If you want to find the area of the surface you parametrize the surface such as $ \vec{S} (x,t) $. Then the surface element is $ || \frac{\partial{ \vec{S}}}{\partial{x}} \times \frac{\partial{\vec{S}}}{\partial{t} }|| dx dt $ and we integrate that. Νow I have a few questions concerning that.


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*If, say, I parametrize the surface using the azimuth $φ$ and the height $z$  then will the surface element be the same as that of a surface of constant radial distance, substituting $ρ$ as a function of $z$ and $φ$ ? $$dS = ρ(z,φ)dzdφ =^? || \frac{\partial{ \vec{S}}}{\partial{z}} \times \frac{\partial{\vec{S}}}{\partial{φ} }|| dz dφ $$ 
I did some "experiments" on a cone and it looks like it's not, but why?

*Is it always possible to parametrize the surface using 2 cartesian coordinates $\vec{S} (x,y)$ (or x and z, or y and z)? What about using cylindrical coordinates?  
 A: For your first question, I will not answer it directly, but by analogy I think this will give enough intuition to explain why it is not as simple as what you ask. Essentially what you are asking is: is the third variable (as a function of the first two) equal to the length of the normal vector (as a function of the parameters)?
To see why it's not so simple, let's look at the case of Cartesian coordinates (instead of the azimuth coordinate you are using). Let's consider the simple situation that the surface is the graph of a function: $z = f(x,y)$. Then the parametric function for the surface is just $\vec{S}(x,y) = (x,y,f(x,y))$. Now your question in this case boils down to: is $f(x,y)$ always equal to $\left| \frac{\partial S}{\partial x} \times \frac{\partial S}{\partial y}\right| $? The latter is equal to $\sqrt{1 + \frac{\partial f}{\partial x}^2 + \frac{\partial f}{\partial y}^2}$, which you certainly don't always expect to be equal to $f(x,y)$ itself.
For your second question: no, it is not always possible to parameterize an entire surface using only Cartesian coordinates. For a simple example, consider a sphere. You cannot do it. You have to split it into two surfaces and parameterize the two hemispheres separately.
