By David V. Widder

Vintage textual content leads from common calculus into extra theoretic difficulties. distinct process with definitions, theorems, proofs, examples and workouts. themes contain partial differentiation, vectors, differential geometry, Stieltjes necessary, endless sequence, gamma functionality, Fourier sequence, Laplace rework, even more. various graded routines with chosen solutions.

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**Extra resources for Advanced Calculus**

**Sample text**

Find . This is a special case of equation (2) where F(x, u) = f(x, u) − u. EXAMPLE C. u = f(g(x, u), h(y, u)). Find . 3 HIGHER DERIVATIVES One may compute the higher derivatives of functions defined implicitly. For example, let us compute for equation (2). We have only to differentiate both sides of equation (3), and to remember that the arguments on the right are x and y and that y itself is the function of x defined by equation (1). Then But y′ is given by equation (3), so that In like manner, we could compute the higher derivatives for Examples B and C.

Type I. 2: Limit test for divergence, �3. Type I. 2: Sufficient conditions for conditional convergence, �4. 5: Oscillating integrands, �5. 4: Combinations of integrals, �6. 1: The Weierstrass M-test, �7. 4: Application to Taylor’s formula, �8. 3: Differentiation, �9. 3: Other methods of summability, �10. 3: The Minkowski inequality, 11 The Gamma Function. Evaluation of Definite Integrals �1. 4: Other expressions for Γ(x), �2. 4: Wallis’s product, �3. 3: The method of infinite series, �4. 5: An infinite product for Γ(x), 12 Fourier Series �1.

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. 2: The principal result, �8. 2: The inversion formula, �9. 3: Illustrations, �10. 3: The Stieltjes transform, 14 Applications of the Laplace Transform �1. 2: Integrands that are determining functions, �2. 3: Equations of higher order, �3. 4: Solution of the problem, �4.