Download Advanced Calculus by David V. Widder PDF

By David V. Widder

ISBN-10: 0486661032

ISBN-13: 9780486661032

Vintage textual content leads from ordinary calculus into extra theoretic difficulties. specified method with definitions, theorems, proofs, examples and workouts. subject matters contain partial differentiation, vectors, differential geometry, Stieltjes crucial, endless sequence, gamma functionality, Fourier sequence, Laplace remodel, even more. a variety of graded routines with chosen solutions.

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3: Equations of higher order, �3. 4: Solution of the problem, �4. 3: Uniqueness of solution, �5. 4: Solution of difference equations, �6. 3: The plucked string, Selected Answers Index of Symbols Index 1 Partial Differentiation �1. Introduction We shall be dealing in this chapter with real functions of several real variables, such as u = f(x, y), u = f(x, y, z), etc. In these examples the variables x, y, z, … are called the independent variables or arguments of the function, u is the dependent variable or the value of the function.

3 A BASIC MEAN-VALUE THEOREM We are now able to establish a result of fundamental importance in the theory of partial differentiation. It may be considered analogous to Theorem 2, the law of the mean for functions of a single variable. We shall the letter D to indicate a domain. Set and rewrite it as follows: Δf = [f(a + Δx, b) − f(a, b)] + [f(a + Δx, b + Δy) − f(a + Δx, b)]. Here we have added and subtracted f(a + Δx, b) on the right-hand side of equation (1). Now apply Theorem 2 to the function f(x, b) of the single variable x.

4: Wallis’s product, �3. 3: The method of infinite series, �4. 5: An infinite product for Γ(x), 12 Fourier Series �1. 3: Further examples of Fourier series, �2. 3: Abbreviations, �3. 4: The convergence theorem, �4. 3: Applications, �5. 6: Special cases, �6. 2: Fejér’s theorem. 3: Uniformity, �7. 5: Uniqueness, �8. 5: Fourier transform, 13 The Laplace Transform �1. 2: Definitions, �2. 3: Examples, �3. 3: Differentiation of generating functions, �4. 5: Illustrations, �5. 3: Application, �6. 2: Transforms of the functions, �7.

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