By David V. Widder

Vintage textual content leads from easy calculus into extra theoretic difficulties. particular technique with definitions, theorems, proofs, examples and workouts. subject matters comprise partial differentiation, vectors, differential geometry, Stieltjes essential, endless sequence, gamma functionality, Fourier sequence, Laplace rework, even more. quite a few graded workouts with chosen solutions.

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2: Comparison tests, �2. 3: Maclaurin’s integral test, �3. Absolute Convergence. 2: Leibniz’s theorem on alternating series, �4. 2: Limit test for divergence, �5. 3: Relation to absolute convergence, �6. 3: Differentiation of series, �7. 4: Other methods of summability, �8. 3: Partial summation, �9. 3: Abel’s theorem, 10 Convergence of Improper Integrals �1. 4: Absolute convergence, �2. Type I. 2: Limit test for divergence, �3. Type I. 2: Sufficient conditions for conditional convergence, �4. 5: Oscillating integrands, �5.

3: Existence of the integral, �2. 3: Volume of a solid, �3. 2: Illustrations, �4. 3: Illustrations, �5. 2: Polar coordinates, �6. 3: Moments of inertia, �7. 5: Attraction, �8. 3: Applications, �9. 2: Spherical coordinates, �10. 4: Area, 7 Line and Surface Integrals �1. 3: Work, �2. 4: Area, �3. 4: Multiply connected regions, �4. 3: Extensions, �5. 5: An auxiliary result, �6. 5: Vector considerations, 8 Limits and Indeterminate Forms �1. 3: L’Hospital’s rule, �2. 1: L’Hospital’s rule, �3. 3: The forms 00, 0∞, ∞0, ∞∞, 1∞, �4.

For, we have only to set and proceed as before. For example, Find . It is more convenient to differentiate the equations directly than to apply the above formulas. In this simple case, we may check by obtaining explicitly the inverse transformation To find , for example, we must solve for . This will be possible if EXERCISES (6) 1. If compute . 2. In the previous exercise compute Formulate a problem to which this is the answer. 3. 3. 4. In Example B complete the computation of . 5. If find by use of Jacobians and then by using the explicit inverse of the transformation.