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Table of Contents:
1. Partial Differentiation -- 1. Introduction -- 2. Functions of One Variable -- 3. Functions of Several Variables -- 4. Homogeneous Functions. Higher Derivatives -- 5. Implicit Functions -- 6. Simultaneous Equations. Jacobians -- 7. Dependent and Independent Variables -- 8. Differentials. Directional Derivatives -- 9. Taylor's Theorem -- 10. Jacobians -- 11. Equality of Cross Derivatives -- 12. Implicit Functions -- 2. Vectors -- 1. Introduction -- 2. Solid Analytic Geometry -- 3. Space Curves -- 4. Surfaces -- 5. A Symbolic Vector -- 6. Invariants -- 3. Differential Geometry -- 1. Arc Length of a Space Curve -- 2. Osculating Plane -- 3. Curvature and Torsion -- 4. Frenet-Serret Formulas -- 5. Surface Theory -- 6. Fundamental Differential Forms -- 7. Mercator Maps -- 4. Applications of Partial Differentiation -- 1. Maxima and minima -- 2. Functions of Two Variables -- 3. Sufficient Conditions -- 4. Functions of Three Variables -- 5. Lagrange's Multipliers -- 6. Families of Plane Curves -- 7. Families of Surfaces -- 5. Stieltjes Integral -- 1. Introduction -- 2. Properties of the Integral -- 3. Integration by Parts -- 4. Laws of the Mean -- 5. Physical Applications -- 6. Continuous Functions -- 7. Existence of Stieltjes Integrals -- 6. Multiple Integrals -- 1. Introduction -- 2. Properties of Double Integrals -- 3. Evaluation of Double Integrals -- 4. Polar Coordinates -- 5. Change in Order of Integration -- 6. Applications -- 7. Further Applications -- 8. Triple Integrals -- 9. Other Coordinates -- 10. Existence of Double Integrals -- 7. Line and Surface Integrals -- 1. Introduction -- 2. Green's Theorem -- 3. Application -- 4. Surface Integrals -- 5. Change of Variable in Multiple Integrals -- 6. Line Integrals in Space -- 8. Limits and Indeterminate Forms -- 1. The Indeterminate Form 0/0 -- 2. The Indeterminate Form -- 3. Other Indeterminate Forms -- 4. Other Methods. Orders of Infinity -- 5. Superior and Inferior Limits -- 9. Infinite Series -- 1. Convergence of Series. Comparison Tests -- 2. Convergence Tests -- 3. Absolute Convergence. Altering Series -- 4. Limit Tests -- 5. Uniform Convergence -- 6. Applications -- 7. Divergent Series -- 8. Miscellaneous Methods -- 9. Power Series -- 10. Convergence of Improper Integrals -- 1. Introduction -- 2. Type I. Limit Tests -- 3. Type I. Conditional Convergence -- 4. Type III -- 5. Combination of Types -- 6. Uniform Convergence -- 7. Properties of Proper Integrals -- 8. Application of Uniform Convergence -- 9. Divergent Integrals -- 10. Integral Inequalities -- 11. The Gamma Function. Evaluation of Definite Integrals -- 1. Introduction -- 2. The Beta Function -- 3. Evaluation of Definite Integrals -- 4. Stirling's Formula -- 12. Fourier Series -- 1. Introduction -- 2. Several Classes of Functions -- 3. Convergence of a Fourier Series to Its Defining Function -- 4. Extensions and Applications -- 5. Vibrating String -- 6. Summability of Fourier Series -- 7. Applications -- 8. Fourier Integral -- 13. The Laplace Transform -- 1. Introduction -- 2. Region of Convergence -- 3. Absolute and Uniform Convergence -- 4. Operational Properties of the Transform -- 5. Resultant -- 6. Tables of Transforms -- 7. Uniqueness -- 8. Inversion -- 9. Representation -- 10. Related Transforms -- 14. Applications of the Laplace Transform -- 1. Introduction -- 2. Linear Differential Equation -- 3. The General Homogeneous Case -- 4. The Nonhomogeneous Case -- 5. Difference Equations -- 6. Partial Differential Equations -- Selected Answers -- Index of Symbols -- Index.
Brief Description:
Classic text offers exceptionally precise coverage of partial differentiation, vectors, differential geometry, Stieltjes integral, infinite series, gamma function, Fourier series, Laplace transform, much more. Includes exercises and selected answers.
Marc Notes:
Reprint. Originally published: Englewood Cliffs, N.J.: Prentice-Hall, 1961. Originally published in series: Prentice-Hall mathematics series.;Includes indexes.
Publisher Marketing:
This classic text by a distinguished mathematician and former Professor of Mathematics at Harvard University, leads students familiar with elementary calculus into confronting and solving more theoretical problems of advanced calculus. In his preface to the first edition, Professor Widder also recommends various ways the book may be used as a text in both applied mathematics and engineering.
Believing that clarity of exposition depends largely on precision of statement, the author has taken pains to state exactly what is to be proved in every case. Each section consists of definitions, theorems, proofs, examples and exercises. An effort has been made to make the statement of each theorem so concise that the student can see at a glance the essential hypotheses and conclusions.
For this second edition, the author has improved the treatment of Stieltjes integrals to make it more useful to the reader less than familiar with the basic facts about the
Riemann integral. In addition the material on series has been augmented by the inclusion of the method of partial summation of the Schwarz-Holder inequalities, and of additional results about power series. Carefully selected exercises, graded in difficulty, are found in abundance throughout the book; answers to many of them are contained in a final section.
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