Advanced Calculus

Klappentext
The Fifth Edition of this leading text offers substantial training in vectors and matrices, vector analysis, and partial differential equations. Vectors are introduced at the outset and serve at many points to indicate geometrical and physical significance of mathematical relations. Numerical methods are touched upon at various points, because of their practical value and the insights they give about theory.

Zusammenfassung
The fifth edition of this text offers substantial training in vectors and matrices, vector analysis, and partial differential equations.

Inhalt


1. Vectors and Matrices.
Introduction. Vectors in Space. Linear Independence Lines and Planes. Determinants. Simultaneous Linear Equations. Matrices. Addition of Matrices Scalar Times Matrix. Multiplication of Matrices. Inverse of a Square Matrix. Gaussian Elimination. *Eigenvalues of a Square Matrix. *The Transpose. *Orthogonal Matrices. Analytic Geometry and Vectors n-Dimensional Space. *Axioms for Vn. Linear Mappings. *Subspaces Rank of a Matrix. *Other Vector Spaces.

2. Differential Calculus of Functions of Several Variables.
Functions of Several Variables. Domains and Regions. Functional Notation Level Curves and Level Surfaces. Limits and Continuity. Partial Derivatives. Total Differential Fundamental Lemma. Differential of Functions of n Variables The Jacobian Matrix. Derivatives and Differentials of Composite Functions. The General Chain Rule. Implicit Functions. *Proof of a Case of the Implicit Function Theorem. Inverse Functions Curvilinear Coordinates. Geometrical Applications. The Directional Derivative. Partial Derivatives of Higher Order. Higher Derivatives of Composite Functions. The Laplacian in Polar, Cylindrical, and Spherical Coordinates. Higher Derivatives of Implicit Functions. Mixima and Minima of Functions of Several Variables. *Extrema for Functions with Side Conditions Lagrange Multipliers. *Maxima and Minima of Quadratic Forms on the Unit Sphere. *Functional Dependence. *Real Variable Theory Theorem on Maximum and Minimum.

3. Vector Differential Calculus.
Introduction. Vector Fields and Scalar Fields. The Gradient Field. The Divergence of a Vector Field. The Curl of a Vector Field. Combined Operations. *Curvilinear Coordinates in Space. Orthogonal Coordinates. *Vector Operations in Orthogonal Curvilinear Coordinates. *Tensors. *Tensors on a Surface or Hypersurface. *Alternating Tensors. Exterior Product.

4. Integral Calculus of Functions of Several Variables.
The Definite Integral. Numerical Evaluation of Indefinite Integrals. Elliptic Integrals. Double Integrals. Triple Integrals and Multiple Integrals in General. Integrals of Vector Functions. Change of Variables in Integrals. Arc Length and Surface Area. Improper Multiple Integrals. Integrals depending on a Parameter—Leibnitz's Rule. *Uniform Continuity. Existence of the Riemann Integral. *Theory of Double Integrals.

5. Vector Integral Calculus.
Two-Dimensional Theory. Introduction. Line Integrals in the Plane. Integrals with Respect to Arc Length. Basic Properties of Line Integrals. Line Integrals as Integrals of Vectors. Green's Theorem. Independence of Path. Simply Connected Domains. Extension of Results to Multiply Connected Domains. Three-Dimensional Theory and Applications. Line Integrals in Space. Surfaces in Space. Orientability. Surface Integrals. The Divergence Theorem. Stokes's Theorem. Integrals Independent of Path.

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