Advanced Calculus (Solutions Manual) by Gerald B. Folland

By Gerald B. Folland

This is often the total ideas handbook to Gerald Folland's Advanced Calculus.

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4. 29 € € € € ✁ ✟ ✍ € ✁ ✜ ➉ € ✄ € ❃ ✆ ✟ ✁ ✁ ❖ ✄ ❃ ➇ ✒ ✁❬☞✎✏✦✍ ✁✾✒ ✁ ✄✝❃ ✟ ✆ ✟ ✝ ✜ ✓ ✎ ✌ ✌ ☎ ✌ ✵ ✌ ➈ The✒ volume is . The -moment is ✝ ✝ ✝ ✒ € € € € € € ✄ ✆ ✟ ✆ ✟ ✟ ✁ ✻ ✸ ✜ ✺ ✼ ✁ ✄ ✼ ✙ ✔ ✕ ➉ ✁ ✭ ✜ ✍ ✩ ✁ ✪ ➉ ➁ ✒ ✓ . The -moment vanishes by symmetry. ✝ ✎ ✆ ✟ ✝ ✝ ✵ ✒ ✌ ✌ € ✌☎✵ € ➈ ✵ ✎ ✆ ✒ ✟ ➈ ✜ € € ✪ ✆ ✟✒ ✁ ✁ ✄✖❃ ☞✎✏❬✍ ✁ ✟ ✒ ✁ ✁ ✄✖❃ € ✁ ✟ ✍ € ✁ ✪✮➉ € ✄ € ❃ ➇❢✄ ✏✓✽▲❃ ✟ ✪ ✪ ✝ . Thus , The -moment is ✝ ✎ ✆ ✟ ✝ ✝ ✌ ✌ ✾✵ ✌ ✌ ➈✟ ➁✉✄❈✰ ❧✄ ✜ ✪ . , and ✄ ✟✝✆ € ✙ ✛ ✍ ✁ ✟ ✁ ✁ ✥✄❉❆ ✛ ✲✍ ✁ ✟ ✄ The top and bottom hemispheres , so ☎ ✝ ✝ ✌ ☎ ✌ ✵ ✒ € ✒ are given by ✍ ✪ ❃❁☞ ✍ ✁ ✟ ✒ ✜ ✟ ✄ ✪ ❃❁☞ ★✑✍⑩✞ ✜ ✟ ✒ ✜ ✜ .

Is sufficiently fine. It follows that this integral equals (b) ✝ ☞❊❛ ✒ ✟ ✡ ☎ ☞❊❛ ❯ ✒ ✟ ❊☞ ❛ ❯ ✍◗❛ ❯ ✎ € ✒ for some ❯ ❛ ❨➂ ❛ ❯ ✐ ❛ ❯ ✎ ✗€ ➂❨❛ ❯ ➉ ✌❢✄ ✏ ✌❢✄ ✁ ➈ . 1, the sum of these quantities from to , which ✆ ☞❊❛❜✒✔✟☛✡ ☎ ☞❊❛❨✒✔✟ ❛❋✄ ☞✄ ✒ ☛ ☞❊❛❨✒✗ ❛ , can be made as close to ✝✠✞ ✠ ✝✟✞ ✌ as we wish by taking ✙ ✞❀☞ ✄✲✒ ✌ 9. By the mean value theorem, the displayed expression equals ✌ ✌ ✝ ✝✄✂ ✝ ✌ ✜ . Chapter 5. Line and Surface Integrals; Vector Analysis 36 ✟ ✟ ✎☞ ✏✗✰✠➇❫✡✘❂▲➁❬✒✮✍ ☞✎✏✗✰✠➇❴✡✘✙▲➁❬✒ ➉ ➇ ➁⑥✄ ➈ ✌ ✌ ✝ ✗ ☞ ✞▲➇ ✟ ✼✄✔✖✕❤➁ ✟ ✒ ☞❊➇ ✟ ✡ ✏✗✰✠➇❍➁❬✡✭➁ ✟ ✒ ➇❴✡ ☞ ❂▲➇ ✟ ✡✘❂▲➇❍➁❬✒ ➁☎➉❴✄ ✌ (c) ✙✦✧✩❅✑✝✁ ✄✳➈ ✏✱✙ ✄ ☎ ☞ ✙▲➇ ✻➁✂✸✻✺✜✼❍➁ ✟ ✒❑✄⑨❅▲➇ ✟ ➁✂✸✻✺✜✼✫➁ ✟ ✌ ✄ ✜ (d) ❽ theorem vanishes.

So ✓ ✓ ✌ ✌ ✌ ✄✕✙▲➇❝✡❢➁ ☞ ➂ ✒❨✽ ☞❊➇✬➂❨➁✫✒❤✄ ✠ ➇★✄ € ☞ ✡★✞ ✒ ➁s✄ € ☞✎✍❀✙ ✡ ✒ ✏ ✁ ✏ 9. ✰ The ✰sq given q € ✄ € by € € ✁ € ✄ ✁ -plane parallelogram in the -plane becomes the rectangle in the given , ✟ ✜ ✏✱❂ ✙ ☞✎✏✗✰✾✒✻☞✎✏✱❂✜✒ ➇ ➇ ➇ ➁❦✄ ☞ ✡⑤✞ ✒ ✄ ✪ ✪ ✄ , . So the area of is ✏ ✝ ✁ ✄✟✄ ✌✎✁ ✌✔ € ✄ € ☞✎✍✤✙ is ✝ ✡ ✝ ☛ ✒ ✌ ✌ ✄✝✍✌ ✄✝ ✜ ✄ ✏ ➁ ➁ ➇ ➁⑥✄ ; the€ -moment ❋ ➇ ✄ ➁ ✝ ✄ ✍ € €✪ . ) ✁ . The center of is clearly , so the center of is ✪ ✄ ➇⑩✡♠➁ ✄ ➇❲✍⑨➁ ☞ ➂ ✒❨✽ ☞❊➇✬➂❨➁✫✒t✄ ✍✤✙ ☞❊➇⑩✡♠➁✫✒ ✪ ☞❊➇✥✍⑨➁❬✒ ✎ ✄ ✄ ✁ ✁ ❽ ✁ ❽ ☛✟ ✌ 10.

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