Elementary Topics in Differential GeometrySpringer Science & Business Media, 27 okt. 1994 - 256 sidor In the past decade there has been a significant change in the freshman/ sophomore mathematics curriculum as taught at many, if not most, of our colleges. This has been brought about by the introduction of linear algebra into the curriculum at the sophomore level. The advantages of using linear algebra both in the teaching of differential equations and in the teaching of multivariate calculus are by now widely recognized. Several textbooks adopting this point of view are now available and have been widely adopted. Students completing the sophomore year now have a fair preliminary under standing of spaces of many dimensions. It should be apparent that courses on the junior level should draw upon and reinforce the concepts and skills learned during the previous year. Unfortunately, in differential geometry at least, this is usually not the case. Textbooks directed to students at this level generally restrict attention to 2-dimensional surfaces in 3-space rather than to surfaces of arbitrary dimension. Although most of the recent books do use linear algebra, it is only the algebra of ~3. The student's preliminary understanding of higher dimensions is not cultivated. |
Innehåll
| 1 | |
| 16 | |
| 31 | |
| 86 | |
Chapter 13 | 101 |
Chapter 8 | 115 |
Chapter 15 | 128 |
Chapter 17 | 144 |
Chapter 19 | 168 |
Chapter 20 | 182 |
Chapter 22 | 217 |
Chapter 24 | 231 |
Bibliography | 245 |
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attains called Chapter choice circle compact compute connected consistent constant containing continuous convex coordinate vector fields critical point curvature cylinder defined definite denned denote derivative determinant differential direction domain dot product equal equation Example Exercise exists fact Figure focal follows formula geodesic geometry given global grad gradient Hence Image integral curve interval isometry Lemma length level set linear matrix maximal maximum metric negative normal Note obtained open set oriented n-surface orthogonal orthonormal basis parallel parametrized curve parametrized n-surface particular plane curve positive principal curvatures Proof regular Remark respect restriction Show singular smooth function smooth map smooth vector field Suppose surface tangent space tangent vector field Theorem unique unit unit speed unit vector v e Sp variation volume zero
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Sida 119 - X be the vector field along <p whose ith component is ( - 1 )" + ' + ' times the determinant of the matrix obtained by deleting the ith column from the matrix where the E; are the coordinate vector fields of <p.
Sida 49 - That this is indeed the case can be seen from the following argument.
Sida 26 - Ln set if each pair of points in S can be joined by a polygonal arc in S having at most n segments.
Sida 11 - Thus, x — a cos T + b sin T, — = — a sin T + b cos T + =£ cos T + -£- sin T.
Sida 8 - R is smooth if all its partial derivatives of all orders exist and are continuous. A function / : U — ^ R* is smooth if each component function fi : U -> R (f(p) = (A(p), /2(p), . . . , A(p)) for p € U) is smooth.
Sida 154 - U (£) is equal to (— 1 )'+J times the determinant of the matrix obtained by deleting the /th row and the ith column in t/(£) (this is Cramer's rule).
Sida 66 - The radius of the circle of curvature is called the radius of curvature of the curve for the point. 368. What is the expression for the radius of curvature? gl [1 + (yQT Д - К - у...
Sida 8 - Rn+ 1 is a function which assigns to each point of U a vector at that point. Thus...
Sida 88 - J(y) < 0 for all v =f= 0, negative aejimte n &yv) < u tor an v =po, definite if it is either positive or negative definite, indefinite if it is neither positive...