Elementary topics in differential geometry - download pdf or read online
By J. A. Thorpe
Long ago decade there was an important switch within the freshman/ sophomore arithmetic curriculum as taught at many, if now not so much, of our schools. This has been led to via the advent of linear algebra into the curriculum on the sophomore point. some great benefits of utilizing linear algebra either within the instructing of differential equations and within the instructing of multivariate calculus are by means of now widely known. numerous textbooks adopting this standpoint are actually on hand and feature been generally 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 toughen the innovations and talents realized throughout the earlier 12 months. regrettably, in differential geometry no less than, this is often frequently now not the case. Textbooks directed to scholars at this point commonly limit recognition to 2-dimensional surfaces in 3-space instead of to surfaces of arbitrary measurement. even though lots of the fresh books do use linear algebra, it is just the algebra of ~3. The student's initial figuring out of upper dimensions isn't cultivated
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Extra info for Elementary topics in differential geometry
Each vector field X along ex is of the form X(t) = (ex(t), Xt(t), ... , X,,+t(t)) where each component Xi is a function along ex. R is smooth. The derivative of a smooth vector field X along ex is the vector field X along ex defined by V( dX t ( t~ ... , -;tt dX n+ 1 (t). ) At) = ( ex(t), dt X(t) measures the rate of change of the vector part (Xt(t), ... , X,,+t(t)) of X(t) along ex. 2). 2 The acceleration Ci(to) is the derivative at to ofthe velocity vector field~. 4) that differentiation of vector fields along parametrized curves has the following properties.
Show that if an n-surface S is represented both asf-l(e) and as g-l(d) where Vf(p) =F 0 and Vg(p) =F 0 for all peS, then for each peS, Vf(p) = AVg(p) for some real number A =F O. 4. Sketch the graph of the function /: Rl .... , Xl) - X~ - 3xi Xl' [Hint: First find the level setf-l(O~ In what region of the plane isf> O? ] The 2-surface graph (f) is called a monkey saddle. 5. 6. Sketch the cylinder e)Ver the graph off(x) = sin x. 7. Verify that a surface of revolution (Example S) is a 2-surface. 8.
A vector field X along a I p ! 2 (b) Euclidean parallelism in IR 2 : (a) parallel vectors; (b) a parallel vector field. 2(b)). Thus X is Euclidean parallel along ex if and only if X = o. Given an n-surface S in IRn + 1 and a parametrized curve ex: I -+ S, a smooth vector field X tangent to S along ex is said to be Levi-Civita parallel, or simply parallel, if X' = O. Intuitively, X is parallel along ex if X is a constant vector field along ex, as seen from S. 47 8 Parallel Transport Note that Levi-Civita parallelism has the following properties: (i) If X is parallel along ex, then X has constant length, since ~ IIXI1 2 = ~ (X • X) = 2X' • X = O.
Elementary topics in differential geometry by J. A. Thorpe