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The FrenetSerret equations are a convenient framework for analyzing curvature. In these equations T(s) represents the unit tangent to the curve as a function of path length s. N(s) is the unit normal to the curve and the vector cross product of T(s) and N(s) is the unit binormal B(s). The remarkable thing is that the derivatives of these unit vectors can be expressed neatly in terms of the unit vectors themselves. In particular,
where k(s) is the curvature. For three dimensional curves
where τ is called the torsion coefficient.
In matrix form these equations are:
 dT/ds    0  k(s)  0    T(s)   
 dN/ds   =   k(s)  0  τ(s)    N(s)  
 dB/ds    0  τ(s)  0    B(s)  
For a curve in a plane B(s) is constant so dB/ds=0 and thus the torsion coefficient τ must be zero. Therefore for a curve in a plane the equations reduce to:
(To be continued.)
There are several differenct concepts of the curvature of a surface. The most important is the one called Gaussian curvature. Gaussian curvature may be defined in several different but equivalent ways. One elegant way is in terms of a Shape Operator that expresses the rate of change of the unit normal vector to a surface with respect to vectors in the tangent plane to the surface. The Gaussian curvature is the determinant of this shape operator.
Another very elegant method of defining the Gaussian curvature of a surface is in terms of differential forms. The defining equation for Gaussian curvature in these terms is the very abstruse equation, that will have to be explained in detail later,
The easy part of the explanation is that K is the Gaussian curvature. A frame field of unit vectors (E_{1}(p), E_{2}(p), E_{3}(p)) is presumed in which E_{3}(p) is the unit normal to the surface at point p of the surface. E_{1}(p) and E_{2}(p) are therefore necessarily unit vectors in the plane tangent to the surface at point p. ζ_{1}(p) and ζ_{2}(p) are the dual 1forms of E_{1}(p) and E_{1}(p). A 1form is a linear functional defined on the set of tangent vectors to E^{3}, three dimensional space. In particular,
The symbol ^ stands for the wedge product of 1forms. Finally the matrix {ω_{i,j} }is the connection form for the frame field of the E_{i}(p)'s. The components of the connection form are 1forms and the differential of one of them is a 2form.
Let U_{i} be the unit vector in the direction of increasing Cartesian coordinate x_{i}. The Cartesian coordinates expressed in terms of spherical coordinates are:
The frame field for a sphere, the unit vectors in the directions of increasing φ, θ and ρ, is then given by
The unit vector in the direction of increaseing ρ is the unit normal vector for the sphere.
The dual 1forms are then
The wedge product of these dual 1forms is
where dφ(v) is equal to the component of the vector v in the direction of increasing φ and likewise for dθ(v).
The relevant component of the connection form for the sphere is
Comparing dω_{1,2}(v) and ζ_{1}^ζ_{}(v) we see that the Gaussian curvature of a sphere has to be 1/ρ^{2}.
For a surface given in terms of two parameters u and v the dual 1forms are
where f and g are functions of the point on the surface.
The formula for Gaussian curvature then implies
For the sphere ω_{1,2} is equal to sin(φ)dθ so the integral for φ from π/2 to φ_{0} and θ from 0 to 2π is then 2π[1sin(φ_{0})], which is known from other techniques.
For more on this topic see the GaussBonnet Theorem.
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