Download PDF by Gerald B. Folland: Advanced Calculus (Solutions Manual)

By Gerald B. Folland

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Hence, the bounded the integrals of these difference € quotients, which are ✘ ☞❊✁✿➂❨✢●✒ ☛✈✢ ☞❊✁❢✡ by ✢ ✒✿✍ ☞❊✁✬✒ ➉❊✽ ☎ ❯ convergence theorem, ✧✗✧✗✧ ☞❊✁✬✒ ❯ , converge to ✝ , which is therefore ❽ . 6 Improper Integrals ☎ ✁ ✌ ➇✞✎✢✜ ✒ ✟ . ➇ ✎ € 1. (a) Converges by comparison to . ☎ ✂ (c) Converges by comparison to (b) Diverges by comparison to ✎ ➇ (for example), using the fact that large . (d) ➇ The integrand is bounded in absolute value by ; hence converges absolutely. € 2. ) and , so ➇❴✽ ✛ ✏✯✍✇➇ ✟ ☛✳✏✓✽ ✆ ✙✫☞✎✏✤✍✇➇❴✒ ➇ for near 1.

However, ➁ integrable ☞❊➇❄➂❨➁ ✒ on ❷ because it is✰✣unbounded ➂✱✏✮➉ ✰ ➁ ➈ for fixed , ❵ is continuous on except at the three points , , and and is in ➁ ✎ ✟ ✰✣➂✱✏✮➉ ☞❊➇ ➂❨➁✫✒ ✰✣➂✱bounded ✏✮➉ absolute value by ; hence it is integrable on . Likewise is integrable on . ➈ ❵ ➈ € € € ✗ € € € € (b)€ ✝ ✝ ✝ ☞❊➇✬➂❨➁✫✒ ➇ ♠ ➁ ✄ € ✝ ❊☞ ➇❄❵➂❨➁❬✒ ➁ ✌ ➇★✄ ✌ € ✍ ❵ ✌ ✌ ✝ ➈ ✑ ✝ ✝ ☎➈✝ ➁ ✎ ✟ ➇ ✍ ✗ ➇ ✎ ✟ ❍➇ ➉ ❖ ➁ ✄ ➁ ✎ ✍❖☞✎✍✑✏✲✡r➁ ✎ ✒ ➉ ➁❖✄ ✏ ✌ ✌ ✌ ➈ ✌ , ✝ ✝ € ➇ ✎ ✟ ➁ ✡ ✛ ➁ ✎ ✟ ➁✾➉ ⑥ ➇ ✄ € ✯ ✍ ➇ ✎ € ✕ ✡ ☞✎✍✑✏✂✡◗➇ ✎ € ✒ ➉ ➇❦✳ ✄ ✍ ✏ .

Finally, if , we apply this result with ✝✟✞ ❵ ✍✂☞❊➇❴✒ ✄ ✍❑☞ ✗✗✒ ✝✠✆ ✞ ❵ ✠ 7. Suppose ✠ ✍✂☞ ✗✗✒ ✠ ✠ ✠ ✠ replaced by and ✠ ✠ 2. ✠ ✠ ✠ to get the desired conclusion. 3 Multiple Integrals and Iterated Integrals 1. ✠ € ☎ € ✎ ☎ ✖ ☞❊➇✕✡ ✞▲➁ ✜ ✒ ➁ ➇ ✄ € ➇ ✛ ✏❝✍✇➇➃✟★✡ ✜ ☞✎✏⑩✍ ➇ ✟ ✒ ✟ ➉ ➇ ✄ ✞▲➁ ✜ ✒ ✄ € € ✪ ☛✟ ✌ ✌ ✌ ✌ (a) € ✝ ✝ ✝ ✎ ✝ € ✄ ✝ ✎ ➈ ✒ € ✟ ✍ ☞✎✏❝✍✪➇ ✟ ✒ ✜ ✟ ✡ ✜ ☞❊➇ ✍ ➇ ✜ ✡ ✄ ➇ ✒ ➉ ✎ € ✄ ✪✄ ✪ ✜ ➈ ✜ . ☞❊➇ ✟ ✍ ✛ ➁✫✒ ✄ ✟ ✗ ✟ ✗ ☞❊➇ ✟ ✍ ✛ ➁❬✒ ➇ ➁❢✄ ✟ € ➇ ✜✯✍✪➇ ✛ ➁✾➉ ✗✟ ✗ ➁♣✄ ✟ ☞ ✞ ➁ ✜✤✍⑩✙▲➁ ✜ ✒ ✟ ✍ € ➁✞✓☛✡ ✁ ✜ ✜ (b) ✝ ✝ ✌✟ ✝ ✝ ✖ ✌ ✌ ✝ ➈✜ ✝ ✖ ✌ ➁ ✄✄✒ ✟ ✒ ➁✉✄ ✟ ➁ ✪ ✍ ✪✄ ➁ ✄✄✒ ✟ ✍ ✟ € € ➁ ✏ ✡ ✟ ➁ ✏ ✒ ✟ ➉ ✟ ✄ ✜ ✄✟ ☞ ❂✘✍ ✛ ✙✾✒ ✜ ✏ ✌ ➈✜ .

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Advanced Calculus (Solutions Manual) by Gerald B. Folland


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