Page 136 - Intro to Tensor Calculus

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are known as the Frenet-Serret formulas of differential geometry.

Surfaces and Curvature

Let us examine surfaces in a Cartesian frame of reference and then later we can generalize our results
to other coordinate systems. A surface in Euclidean 3-dimensional space can be defined in several different
ways. Explicitly, z = f(x, y), implicitly, F(x, y, z) = 0 or parametrically by defining a set of parametric
equations of the form
x = x(u, v), y = y(u, v), z = z(u, v)

which contain two independent parameters u, v called surface coordinates. For example, the equations

x = a sin θ cos φ, y = a sin θ sin φ, z = a cos θ

are the parametric equations which define a spherical surface of radius a with parameters u = θ and v = φ.
See for example figure 1.3-20 in section 1.3. By eliminating the parameters u, v one can derive the implicit
form of the surface and by solving for z one obtains the explicit form of the surface. Using the parametric
form of a surface we can define the position vector to a point on the surface which is then represented in
terms of the parameters u, v as

r r (1.5.14)
~ = ~(u, v)= x(u, v) b e 1 + y(u, v) b e 2 + z(u, v) b e 3 .
The coordinates (u, v) are called the curvilinear coordinates of a point on the surface. The functions
x(u, v),y(u, v),z(u, v) are assumed to be real and differentiable such that ∂~ r × ∂~ r 6=0. The curves
∂u ∂v
r
r ~(u, c 2 ) and ~(c 1 ,v) (1.5.15)

with c 1 ,c 2 constants, then define two surface curves called coordinate curves, which intersect at the surface
coordinates (c 1 ,c 2 ). The family of curves defined by equations (1.5.15) with equally spaced constant values
c i ,c i +∆c i ,c i +2∆c i ,... define a surface coordinate grid system. The vectors ∂~ r and ∂~ r evaluated at the
∂u ∂v
surface coordinates (c 1 ,c 2 ) on the surface, are tangent vectors to the coordinate curves through the point
2
1
1
2
3
and are basis vectors for any vector lying in the surface. Letting (x, y, z)=(y ,y ,y )and (u, v)=(u ,u )
and utilizing the summation convention, we can write the position vector in the form
1 2 i 1 2
~ = ~(u ,u )= y (u ,u ) b e i . (1.5.16)
r
r
The tangent vectors to the coordinate curves at a point P can then be represented as the basis vectors
∂~ ∂y i
r
~
E α = = b e i , α =1, 2 (1.5.17)
∂u α ∂u α
where the partial derivatives are to be evaluated at the point P where the coordinate curves on the surface
intersect. From these basis vectors we construct a unit normal vector to the surface at the point P by
r
calculating the cross product of the tangent vector ~ u = ∂~ r and ~ v = ∂~ r . A unit normal is then
r
∂u ∂v
~ ~
r
E 1 × E 2 ~ u × ~r v
b n = bn(u, v)= = (1.5.18)
~
~
r
r
|E 1 × E 2 | |~ u × ~ v |