Definition 93 : A surface is the range of a function I X J → ℜ3, where I and J are intervals on the real axis.
We suppose the function is in each concrete case as often continuously differentiable as is necessary in that case.
We write x(u1,u2) = (x1(u1,u2), x2(u1,u2), x3(u1,u2)).
We can parametrize the same surface in different ways.
We also suppose that in each point x(a,b) the tangents to the parameter lines u1=a and u2=b, so xu1(a,b) and xu2(a,b) are linearly independent, unless we say otherwise.

Explanation 94 : The parameter line u1=a is the curve x(a,u2) with in x(a,b) tangent vector xu2(a,b).

Examples 95 :

i) The sphere with radius R and center m has parametization x(θ,φ) = m + R(sin(θ)cos(φ), sin(θ)sin(φ), cos(θ); so x1(θ,φ) = m1 + R sin(θ)cos(φ), etc.
ii) The surface of revolution x(u,v) = (u cos(v), u sin(v), f(u)), where f is a function of u, comes about by revolving the curve (u, 0, f(u)) around the x3-axis.
iii) The graph of a function f(u,v) of two variables has parametrization (u, v, f(u,v)).

Proposition 96 : Locally, each surface has a parametrization in the form 95 iii), or with other words: we can locally describe it with an equation of the form x3 = f(x1, x2) (or x2 = f(x1, x3) or x1 = f(x2, x3))
Proof : According to the implicit function theorem, u1 and u2 are implicitly given by the equations x1 = x1(u1,u2) and x2 = x2(u1,u2) as functions of x1 and x2. Substitution in x3 = x3(u1,u2) gives the required result.

Explanation 97 : In the proof of proposition 96 we use the third condition of definition 93. We call a point where xu1 and xu2 are linearly dependent a singular point of the parameter representation.
Thus, the points where θ=0,π are singular points of the parametrization of the sphere in 95 i).

Proposition 98 : The tangent plane in x(a,b) has direction vectors xu1(a,b) and xu2(a,b), so the normal on the surface has direction vector xu1xu2(a,b).
Proof : The tangents in x(a,b) to the parameter lines span the tangent plane.

Problem 99 : Describe the parameter lines in the examples 95 i), ii) and iii) using geometrical terms.

Problem 100 : Consider the surface with parametrization x = (u, uv, uv2).

i) Describe the u-lines (v=c) using geometrical terms.
ii) Does a singular point exist? Can this point be regular with respect to an other parametrization?
iii) Give an equation of the tangent plane in a regular point x(u,v).
iv) Prove that the tangent plane along an u-line (in this case a straight line on the conic) is constant.

Problem 101 : Give the equations x3 = f(x1, x2) for the examples 95 i), ii) en iii).

Problem 102 : Give the tangent planes in the examples 95 i), ii) en iii).