By J. A. Thorpe
Long ago decade there was an important swap within the freshman/ sophomore arithmetic curriculum as taught at many, if no longer such a lot, of our schools. This has been led to through the creation of linear algebra into the curriculum on the sophomore point. the benefits of utilizing linear algebra either within the educating of differential equations and within the instructing of multivariate calculus are through now well known. a number of textbooks adopting this perspective at the moment are on hand and feature been broadly followed. scholars finishing the sophomore yr now have a good initial less than status of areas of many dimensions. it's going to be obvious that classes at the junior point should still draw upon and strengthen the thoughts and talents discovered throughout the earlier 12 months. regrettably, in differential geometry at the least, this can be often no longer the case. Textbooks directed to scholars at this point as a rule limit realization to 2-dimensional surfaces in 3-space instead of to surfaces of arbitrary size. even though many of the fresh books do use linear algebra, it is just the algebra of ~3. The student's initial realizing of upper dimensions isn't really cultivated
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Extra resources for Elementary topics in differential geometry
X~+ 1 = 1 is connected if and only if n ~ 1. In this book,· we shall deal almost exclusively with connected n-surfaces. surface,and it is connected. ' 'Theorem 1. Let S c: IRn + 1 be a connected n-surface in IR" + 1. Then there exist on S exactly two smooth unit normal vector fields N 1 and N 2, and N 2 (P) = -N 1(P)for all pe S. PROOF. Letf: U -+ IR and c e IR be such that S = f--l(C) and Vf(p) ::/= 0 for all pES. Then the vector field N 1 on S defined by Vf(P) N 1 (p) = I/ Vf(p)/I ' PES clearly has the required properties, as does the vector field N 2 defined by N 2 (p) = -N 1 (p) for all pES.
N(P) = - v, then N(q) = v where q is obtained similarly, by moving the n-plane in from the opposite direction. More precisely, consider the function g: RII + 1 -+ R defined by g(p) = p. , g(Xh ... , x,,+d = alxl + ... •. , Q,,+l)' The level sets of g are the n-planes parallel toff. Since S is cOmpact, the restriction to S of the function gattains its maximum and its minimum, say at p and q respectively. (P)UN(P) for some l e R. Hence v and N(p)are multiples of one another. Since both have unit length, it follows that N(P) = ±v.
Of/oxn cannot be simultaneously zero· when g(Xh ... , Xn+1) = f(Xh ... , Xn) = c because Vf(Xh"" xn) =F 0 whenever (Xh ... , xn) E f-1(C). 2). 2 The cylinder g-1(1) over the n-sphere: g(X1' ... , Xn + d = xi + ... + x;. EXAMPLE 5. Let C be a curve in 1R2 which lies above the x I-axis. Thus C = f-1(C) for some f: U ~ IR with Vf(p) =1= 0 for all p E C, where U is contained in the upper half plane X2 > O. Define S = g- l(C) where g: U x IR ~ IR by g(Xh X2' X3) = f(Xh (x~ + X~)1/2). 7). Each point p = (a, b) E C generates a circle of points of S, namely the circle in the plane Xl = a consisting of those points (Xh X2, X3) E 1R3 such that Xl = a, x~ + x~ = b2.
Elementary topics in differential geometry by J. A. Thorpe