Calculus on Manifolds
                                             ASolution Manual for Spivak (1965)
                                                          Jianfei Shen
                                      School of Economics, The University of New South Wales
                                                     Sydney, Australia    2010
                           Contents
                           1   Functions on Euclidean Space .....................................   1
                               1.1 NormandInnerProduct.......................................       1
                               1.2 Subsets of Euclidean Space ....................................  6
                               1.3 Functions and Continuity ......................................  9
                           2   Differentiation..................................................... 13
                               2.1 Basic Definitions .............................................. 13
                               2.2 Basic Theorems ............................................... 18
                               2.3 Partial Derivatives ............................................. 26
                               2.4 Derivatives.................................................... 34
                               2.5 Inverse Functions ............................................. 38
                               2.6 Implicit Functions ............................................. 40
                           3   Integration ........................................................ 45
                               3.1 Basic Definitions .............................................. 45
                               3.2 Measure Zero and Content Zero................................ 51
                               3.3 Fubini’s Theorem.............................................. 51
                           4   Integration on Chains ............................................. 55
                               4.1 Algebraic Preliminaries ........................................ 55
                           References ............................................................ 57
                           Index .................................................................. 59
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