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16.R: Chapter 16 Review Exercises

Last updated

Apr 22, 2024
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- 16.8E: Exercises for Section 16.8
- 17: Second-Order Differential Equations

Gilbert Strang & Edwin “Jed” Herman
OpenStax

( \newcommand{\kernel}{\mathrm{null}\,}\)

True or False? Justify your answer with a proof or a counterexample.

1. The vector field $⇀ 𝐅 (𝑥, 𝑦) = 𝑥 2 𝑦 ˆ 𝐢 + 𝑦 2 𝑥 ˆ 𝐣$ is conservative.

Answer: False

2. For vector field $⇀ 𝐅 (𝑥, 𝑦) = 𝑃 (𝑥, 𝑦) ˆ 𝐢 + 𝑄 (𝑥, 𝑦) ˆ 𝐣$ , if $𝑃 𝑦 (𝑥, 𝑦) = 𝑄 𝑧 (𝑥, 𝑦)$ in open region $𝐷$ , then $\int 𝜕 𝐷 𝑃 𝑑 𝑥 + 𝑄 𝑑 𝑦 = 0 .$

3. The divergence of a vector field is a vector field.

Answer: False

4. If $𝑐 𝑢 𝑟 𝑙 ⇀ 𝐅 = ⇀ 𝟎$ , then $⇀ 𝐅$ is a conservative vector field.

Draw the following vector fields.

5. $⇀ 𝐅 (𝑥, 𝑦) = 12 ˆ 𝐢 + 2 𝑥 ˆ 𝐣$

Answer: $A vector field in two dimensions. All quadrants are shown. The arrows are larger the further from the y axis they become. They point up and to the right for positive x values and down and to the right for negative x values. The further from the y axis they are, the steeper the slope they have.$

6. $⇀ 𝐅 (𝑥, 𝑦) = \sqrt 𝑦 ˆ 𝐢 + 3 𝑥 ˆ 𝐣 𝑥 2 + 𝑦 2$

Are the following the vector fields conservative? If so, find the potential function $⇀ 𝐅$ such that $⇀ 𝐅 = ⇀ \nabla 𝑓$ .

7. $⇀ 𝐅 (𝑥, 𝑦) = 𝑦 ˆ 𝐢 + (𝑥 - 2 𝑒 𝑦) ˆ 𝐣$

Answer: Conservative, $𝑓 (𝑥, 𝑦) = 𝑥 𝑦 - 2 𝑒 𝑦$

8. $⇀ 𝐅 (𝑥, 𝑦) = (6 𝑥 𝑦) ˆ 𝐢 + (3 𝑥 2 - 𝑦 𝑒 𝑦) ˆ 𝐣$

9. $⇀ 𝐅 (𝑥, 𝑦) = (2 𝑥 𝑦 + 𝑧 2) ˆ 𝐢 + (𝑥 2 + 2 𝑦 𝑧) ˆ 𝐣 + (2 𝑥 𝑧 + 𝑦 2) ˆ 𝐤$

Answer: Conservative, $𝑓 (𝑥, 𝑦, 𝑧) = 𝑥 2 𝑦 + 𝑦 2 𝑧 + 𝑧 2 𝑥$

10. $⇀ 𝐅 (𝑥, 𝑦, 𝑧) = (𝑒 𝑥 𝑦) ˆ 𝐢 + (𝑒 𝑥 + 𝑧) ˆ 𝐣 + (𝑒 𝑥 + 𝑦 2) ˆ 𝐤$

Evaluate the following integrals.

11. $\int 𝐶 𝑥 2 𝑑 𝑦 + (2 𝑥 - 3 𝑥 𝑦) 𝑑 𝑥$ , along $𝐶 : 𝑦 = 12 𝑥$ from $(0, 0)$ to $(4, 2)$

Answer: $- 163$

12. $\int 𝐶 𝑦 𝑑 𝑥 + 𝑥 𝑦 2 𝑑 𝑦$ , where $𝐶 : 𝑥 = \sqrt 𝑡, 𝑦 = 𝑡 - 1, 0 \leq 𝑡 \leq 1$

13. $\iint 𝑆 𝑥 𝑦 2 𝑑 𝑆,$ where $𝑆$ is the surface $𝑧 = 𝑥 2 - 𝑦, 0 \leq 𝑥 \leq 1, 0 \leq 𝑦 \leq 4$

Answer: $32 \sqrt 2 9 (3 \sqrt 3 - 1)$

Find the divergence and curl for the following vector fields.

14. $⇀ 𝐅 (𝑥, 𝑦, 𝑧) = 3 𝑥 𝑦 𝑧 ˆ 𝐢 + 𝑥 𝑦 𝑒 𝑥 ˆ 𝐣 - 3 𝑥 𝑦 ˆ 𝐤$

15. $⇀ 𝐅 (𝑥, 𝑦, 𝑧) = 𝑒 𝑥 ˆ 𝐢 + 𝑒 𝑥 𝑦 ˆ 𝐣 - 𝑒 𝑥 𝑦 𝑧 ˆ 𝐤$

Answer: Divergence: $𝑒 𝑥 + 𝑥 𝑒 𝑥 𝑦 + 𝑥 𝑦 𝑒 𝑥 𝑦 𝑧$
Curl: $𝑥 𝑧 𝑒 𝑥 𝑦 𝑧 ˆ 𝐢 - 𝑦 𝑧 𝑒 𝑥 𝑦 𝑧 ˆ 𝐣 + 𝑦 𝑒 𝑥 𝑦 ˆ 𝐤$

Use Green’s theorem to evaluate the following integrals.

16. $\int 𝐶 3 𝑥 𝑦 𝑑 𝑥 + 2 𝑥 𝑦 2 𝑑 𝑦$ , where $𝐶$ is a square with vertices $(0, 0), (0, 2), (2, 2)$ and $(2, 0) .$

17. $\oint 𝐶 3 𝑦 𝑑 𝑥 + (𝑥 + 𝑒 𝑦) 𝑑 𝑦$ , where $𝐶$ is a circle centered at the origin with radius $3 .$

Answer: $- 2 𝜋$

Use Stokes’ theorem to evaluate $\iint 𝑆 𝑐 𝑢 𝑟 𝑙 ⇀ 𝐅 \cdot 𝑑 𝑆$ .

18. $⇀ 𝐅 (𝑥, 𝑦, 𝑧) = 𝑦 ˆ 𝐢 - 𝑥 ˆ 𝐣 + 𝑧 ˆ 𝐤$ , where $𝑆$ is the upper half of the unit sphere

19. $⇀ 𝐅 (𝑥, 𝑦, 𝑧) = 𝑦 ˆ 𝐢 + 𝑥 𝑦 𝑧 ˆ 𝐣 - 2 𝑧 𝑥 ˆ 𝐤$ , where $𝑆$ is the upward-facing paraboloid $𝑧 = 𝑥 2 + 𝑦 2$ lying in cylinder $𝑥 2 + 𝑦 2 = 1$

Answer: $- 𝜋$

Use the divergence theorem to evaluate $\iint 𝑆 ⇀ 𝐅 \cdot 𝑑 𝑆$ .

20. $⇀ 𝐅 (𝑥, 𝑦, 𝑧) = (𝑥 3 𝑦) ˆ 𝐢 + (3 𝑦 - 𝑒 𝑥) ˆ 𝐣 + (𝑧 + 𝑥) ˆ 𝐤$ , over cube $𝑆$ defined by $- 1 \leq 𝑥 \leq 1, 0 \leq 𝑦 \leq 2, 0 \leq 𝑧 \leq 2$

21. $⇀ 𝐅 (𝑥, 𝑦, 𝑧) = (2 𝑥 𝑦) ˆ 𝐢 + (- 𝑦 2) ˆ 𝐣 + (2 𝑧 3) ˆ 𝐤$ , where $𝑆$ is bounded by paraboloid $𝑧 = 𝑥 2 + 𝑦 2$ and plane $𝑧 = 2$

Answer: $31 𝜋 / 2$

22. Find the amount of work performed by a 50-kg woman ascending a helical staircase with radius 2 m and height 100 m. The woman completes five revolutions during the climb.

23. Find the total mass of a thin wire in the shape of a semicircle with radius $\sqrt 2$ , and a density function of $𝜌 (𝑥, 𝑦) = 𝑦 + 𝑥 2$ .

Answer: $\sqrt 2 (2 + 𝜋)$

24. Find the total mass of a thin sheet in the shape of a hemisphere with radius $2$ for $𝑧 \geq 0$ with a density function $𝜌 (𝑥, 𝑦, 𝑧) = 𝑥 + 𝑦 + 𝑧$ .

25. Use the divergence theorem to compute the value of the flux integral over the unit sphere with $⇀ 𝐅 (𝑥, 𝑦, 𝑧) = 3 𝑧 ˆ 𝐢 + 2 𝑦 ˆ 𝐣 + 2 𝑥 ˆ 𝐤$ .

Answer: $2 𝜋 / 3$

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

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