GM Week 9

GM Week 9

1. Parametric Surfaces

   • By extension, a surface S in R 3 is parameterised by a continuous map from a region in Ω in R 2 into R 3 , that is r: Ω ⊆ R 2 → R 3 We think of the region Ω getting mapped to the surface in R 3 .
   • In terms of components, we write r(u, v) =(x(u, v), y(u, v), z(u, v)) with (u, v) ∈ Ω We say that S = {r(u, v)|(u, v) ∈ Ω} is a parametric surface and is given parametrically by parameters (u, v) in the region (or domain) Ω.
     • One often draws and analyses surfaces in terms of individual parameterised curves corresponding to the parameters u and v separately. For example, one fixes v to some value v0 and varies u. r(·, v0) is a parameterised curve in S. We call this a u-curve. There is a family of these since the value v0 can be varied. Similarly there is a family of v-curves. These curves are sometimes collectively called grid curves.
   • The parametric representation of a surface S is not unique

2. Examples

   • Unit Sphere: r(θ, φ) = (cos θ sin φ,sin θ sin φ, cos φ)
   • Parametrisation of Hemisphere r(u, v) = (u, v, (1 − u^2 − v^2)^0.5), u^2 + v^2 =< 1
   • Cylinder of Radius R: r(θ, t) = (R cos θ, R sin θ, t) (θ, t) ∈ [0, 2π] × [0, h]
   • Surface of revolution Consider a function f : [a, b] → R +. The graph of f is a curve {(x, y)| x ∈ [a, b], y = f(x) > 0}. Now extend to R 3 and generate a surface by rotating the graph of f about the x axis. This surface is given parametrically by r(x, θ) = (x , f(x) cos θ , f(x) sin θ ) , (x, θ) ∈ [a, b] × [0, 2π]
   • Tours: r(u, v) = (a + b cos u) cos v,(a + b cos u) sin v, b sin u ), where v ∈ [0, 2π] is the angle in the x-y plane measured from the x-axi
   • Graph The graph of f : Ω ⊂ R 2 → R is a surface parametrised by: r(u, v) = u, v, f(u, v) )

3. Tangent Plane and Normal to a Surface

   • ∂r/∂u(u0, v0) is tangent to the u-curve. Since the curve is in S, this vector will be tangent to S at point r(u0, v0). Likewise ∂r/∂v (u0, v0) will be another vector tangent to S at r(u0, v0). These two vectors form a basis for the tangent plane to S at r(u0, v0). That is any point in the plane tangent to S at r(u0, v0) can be represented by a linear combination of these two vectors, plus the point r(u0, v0) itself
     • We can express the position vector of every point of the tangent plane at r(u0, v0) at, where (h, k) ∈ R, :
       • • Frequently one does not express as plane via a parametrisation as above, but in terms of the normal vector
           • • From the normal vector at a point r0 on the surface, one can write an expression for the tangent plane as P = {r ∈ R 3 | n · (r − r0) = 0}

4. Surface Area

   • Given a parameterised surface S, we would like to find the area of the surface
     • • One can show that the area of S is independent of the parametrisation of S.

5. Surface Integrals

   • Given a continuous scalar function f : R 3 → R and a surface S in R 3 , we can evaluate f at a each point on S. Thus we have a real value associated to each point on S and we can integrate this over the surface. Rather than again going through all the arguments of summing up pieces and taking the limit as the number of pieces goes to infinity, we simply state the unsurprising result
     • • In the case where S is a closed surface, one often write ( to indicate specifically that the surface is close)

6. Flux Integrals

   • Given a three-dimensional vector field v: R 3 → R 3 and a surface S in R 3 , one commonly wishes to calculate the flux of v though S . This is defined to be the surface integral, where n is a unit normal vector to S.
     • • Flux integrals are particular surface integrals in which f = v·n. Notice that this f is an example of a function that is only defined on the surface; it has no meaning except in conjunction with the surface. Since v · n is just a scalar, flux integrals are in principle just a particular case of surface integral

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