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DIFFERENTIAL GEOMETRY COURSE

11. *SURFACES*

*Definition 93 :* A surface is the range of a function *I X J → ℜ*^{3}, 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__(u_{1},u_{2}) = (x_{1}(u_{1},u_{2}), x_{2}(u_{1},u_{2}), x_{3}(u_{1},u_{2})).

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 u_{1}=a and u_{2}=b, so __x___{u1}(a,b) and
__x___{u2}(a,b) are linearly independent, unless we say otherwise.

*Explanation 94 :* The parameter line u_{1}=a is the curve __x__(a,u_{2}) with in __x__(a,b) tangent vector __x___{u2}(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 x_{1}(θ,φ) = m_{1} + 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 x_{3}-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
x_{3} = f(x_{1}, x_{2}) (or x_{2} = f(x_{1}, x_{3}) or x_{1} = f(x_{2}, x_{3}))

*Proof :* According to the implicit function theorem, u_{1} and u_{2} are implicitly given by the equations x_{1} = x_{1}(u_{1},u_{2}) and
x_{2} = x_{2}(u_{1},u_{2}) as functions of x_{1} and x_{2}.
Substitution in x_{3} = x_{3}(u_{1},u_{2}) 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 __x___{u1} and __x___{u2}
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 __x___{u1}(a,b) and __x___{u2}(a,b), so the normal
on the surface has direction vector __x___{u1}⊗ __x___{u2}(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, uv^{2}).

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 x_{3} = f(x_{1}, x_{2}) for the examples 95 i), ii) en iii).

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

answers

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