This book explores the standard problem-solving techniques of multivariable mathematics — integrating vector algebra ideas with multivariable calculus and differential equations. Provides many routine, computational exercises illuminating both theory and practice. Offers flexibility in coverage — topics can be covered in a variety of orders, and subsections (which are presented in order of decreasing importance) can be omitted if desired. Provides proofs and includes the definitions and statements of theorems to show how the subject matter can be organized around a few central ideas. Includes new sections on: flow lines and flows; centroids and moments; arc-length and curvature; improper integrals; quadratic surfaces; infinite series—with application to differential equations; and numerical methods. Presents refined method for solving linear systems using exponential matrices.
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DOT PRODUCTS AND CROSS PRODUCTS
CHAPTERS EQUATIONS AND MATRICES
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apply approximation arrow boundary chain rule Chapter coefficients column compute constant continuously differentiable converges coordinate functions curl F curve defined definition determined differentiable function differential equation direction distance domain dot product dx/dt dy/dt eigenvectors entries Example Exercise exponential field F Figure Find first-order fix,y flow lines formula Gauss's theorem geometric given gradient field graph Green's theorem Hence Hint homogeneous identically zero initial conditions intersection interval inverse iterated integral Laplace transform length line integral linear combination linearly independent maximum method multiple nonzero open set parametric representation partial derivatives particular solution path perpendicular point x0 polynomial positive Proof radius real numbers real-valued function rectangle region satisfies scalar Section Show shown in Fig sketch solve Suppose surface tangent vector tank Taylor expansion Theorem trajectory variable vector field velocity