By Michael Spivak

ISBN-10: 0914098713

ISBN-13: 9780914098713

Ebook via Michael Spivak, Spivak, Michael

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**Extra resources for A Comprehensive Introduction to Differential Geometry**

**Example text**

Euclidean metric. 1. Let n = 2. IJ = (g~ = i¢j, while relative to polar coordinates we have: This means that for the curve r =r(t), $ =$(t) b /= f tT 2 2 ('(t) +r d$ 2 (iF) dr. a 2. n = 3. Relative to Canesian coordinates, we have gij = 5U; relative to cynlindrical I, z =r: coordinates r =yl, cI> =

This fact is sometimes also expressed by the formula: where * denotes the vector through which the vector v (or n) is rotated in moving along the curve. The sign indicates the direction (clockwise or counter-elockwise) in which the frame (v, n) is rotated when moving along the curve. The parameter twas always taken to be the natural one. It is now natural to ask how we go about calculating the curvature of a flat cun'e parametrized as ret) = (x(t), y(t)), where t is not the natural parameter? *

A(t)b) = (a. b) for any vectors a and b. Differentiating this identity with respect to t, we obtain the equality (A' (t)a. A(t)b ) + (A (t)a. A '(t)b) = O. When t = O. we obtain (Xa. b) -t (a. Xb) = O. Setting a = ei. b = ejt we come to (Xei • ej) =":" (eit Xej)' Le. ~j = - xii' where X = (x~). Here ei and ej are onhonormal. basic vectors, as required. 6 Space Curves We now proceed to the theory of space curves. For any curve x x(t), Y yet), z z(t) or in terms of the veaors r holds the equalities: = = = =ret) there As in the planar case, we shall first consider the natural parameter 1only, since it is in terms of I that our basic concepts are most conveniently defined.

### A Comprehensive Introduction to Differential Geometry by Michael Spivak

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