Author | Fleming, Wendell. author |
---|---|
Title | Functions of Several Variables [electronic resource] / by Wendell Fleming |
Imprint | New York, NY : Springer New York, 1977 |
Edition | 2nd Edition |
Connect to | http://dx.doi.org/10.1007/978-1-4684-9461-7 |
Descript | XII, 412 p. online resource |
1 Euclidean spaces -- 1.1 The real number system -- 1.2 Euclidean En -- 1.3 Elementary geometry of En -- 1.4 Basic topological notions in En -- *1.5 Convex sets -- 2 Elementary topology of En -- 2.1 Functions -- 2.2 Limits and continuity of transformations -- 2.3 Sequences in En -- 2.4 Bolzano-Weierstrass theorem -- 2.5 Relative neighborhoods, continuous transformations -- 2.6 Topological spaces -- 2.7 Connectedness -- 2.8 Compactness -- 2.9 Metric spaces -- 2.10 Spaces of continuous functions -- *2.11 Noneuclidean norms on En -- 3 Differentiation of real-valued functions -- 3.1 Directional and partial derivatives -- 3.2 Linear functions -- **3.3 Difierentiable functions -- 3.4 Functions of class C(q) -- 3.5 Relative extrema -- *3.6 Convex and concave functions -- 4 Vector-valued functions of several variables -- 4.1 Linear transformations -- 4.2 Affine transformations -- 4.3 Differentiable transformations -- 4.4 Composition -- 4.5 The inverse function theorem -- 4.6 The implicit function theorem -- 4.7 Manifolds -- 4.8 The multiplier rule -- 5 Integration -- 5.1 Intervals -- 5.2 Measure -- 5.3 Integrals over En -- 5.4 Integrals over bounded sets -- 5.5 Iterated integrals -- 5.6 Integrals of continuous functions -- 5.7 Change of measure under affine transformations -- 5.8 Transformation of integrals -- 5.9 Coordinate systems in En -- 5.10 Measurable sets and functions; further properties -- 5.11 Integrals: general definition, convergence theorems -- 5.12 Differentiation under the integral sign -- 5.13 Lp-spaces -- 6 Curves and line integrals -- 6.1 Derivatives -- 6.2 Curves in En -- 6.3 Differential 1-forms -- 6.4 Line integrals -- *6.5 Gradient method -- *6.6 Integrating factors; thermal systems -- 7 Exterior algebra and differential calculus -- 7.1 Covectors and differential forms of degree 2 -- 7.2 Alternating multilinear functions -- 7.3 Multicovectors -- 7.4 Differential forms -- 7.5 Multivectors -- 7.6 Induced linear transformations -- 7.7 Transformation law for differential forms -- 7.8 The adjoint and codifferential -- *7.9 Special results for n = 3 -- *7.10 Integrating factors (continued) -- 8 Integration on manifolds -- 8.1 Regular transformations -- 8.2 Coordinate systems on manifolds -- 8.3 Measure and integration on manifolds -- 8.4 The divergence theorem -- *8.5 Fluid flow -- 8.6 Orientations -- 8.7 Integrals of r-forms -- 8.8 Stokesโs formula -- 8.9 Regular transformations on submanifolds -- 8.10 Closed and exact differential forms -- 8.11 Motion of a particle -- 8.12 Motion of several particles -- Axioms for a vector space -- Mean value theorem; Taylorโs theorem -- Review of Riemann integration -- Monotone functions -- References -- Answers to problems