Advanced Calculus - download pdf or read online

By David V. Widder

ISBN-10: 0486661032

ISBN-13: 9780486661032

Vintage textual content leads from user-friendly calculus into extra theoretic difficulties. particular method with definitions, theorems, proofs, examples and workouts. themes contain partial differentiation, vectors, differential geometry, Stieltjes essential, limitless sequence, gamma functionality, Fourier sequence, Laplace remodel, even more. quite a few graded routines with chosen solutions.

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Example text

The essential feature of the condition is precisely the nonvanishing of the functions which appear in the denominators when computing the first partial derivative. Thus, for equation (1) it is F3 ≠ 0; for equation (2), F2 ≠ 0. For Example B the condition is f2 ≠ 1, and for Example C it is f1g2 + f2h2 − 1 ≠ 0. The student should be careful to insist explicitly on the nonvanishing of every denominator. Observe that it may be possible to solve a given equation for any one of the variables appearing.

Thus, f(x) = (sin x)/x is not continuous at x = 0 in the first instance, since division by zero is undefined. However, if f(0) is defined as 1, f(x) becomes continuous at x = 0. In Example B, f(x) is discontinuous at x = 0 on two counts: f(0) is undefined, and the limit involved does not exist. No choice of definition for f(0) could make f(x) continuous at x = 0. If in Definition 1 “x → a” is replaced by “x → (“x → a−”), f(x) is said to be continuous on the right (left) at x = a. Thus, in Example C, f(x) is continuous on the right at x = 0 if f(0) = 0.

3 THIRD ILLUSTRATION Find if CASE I. The second equation alone is sufficient. CASE II. The first equation alone is sufficient. CASE III. Both equations are necessary. EXERCISES (7) 1. If compute and . 2. In the previous exercise compute (all meanings). 3. If compute . 4. Same problem for . 5. 1. 6. 1, find the two derivatives with respect to the other independent variable. 7. For equations (1) find . 8. Find if 9. Find if 10. For equations (2), enumerate all cases in which both equations are necessary.

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Advanced Calculus by David V. Widder


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