Math 20E: Vector Calculus
Summer Session II, 2015
Guidelines
- Homework is to be turned in to the homework boxes in the basement of APM (map) by the specified time and date.
- Please observe the following neatness guidelines for homework that
you turn in to be graded:
- Use clean, white paper (preferably ruled) that is not torn from a spiral notebook.
- Write your name, PID, and discussion section (e.g. A02) clearly on the front page of your completed assignment.
- Write clearly and legibly.
- Clearly number each solution and present them in numerical order.
Optional Review Problems
Review: Not to be turned in (these topics were covered in Math 20C)
- Section 1.1: (pg 18) 1, 4, 7, 11, 17
- Section 1.2: (pg 29) 3, 7, 12, 22
- Section 1.3: (pg 49) 2b, 5, 11, 15ad, 16b, 30
- Section 1.4: (pg 58) 1, 3ab, 9, 10, 11
- Section 2.1: (pg 85) 2, 9, 10b, 30, 40
- Section 2.2: (pg 103) 2, 6, 9c, 10b, 16
- Section 2.4: (pg 123) 1, 3, 9, 14, 17
- Section 2.6: (pg 142) 2b, 3b, 9b, 10c, 20
- Section 3.1: (pg 156) 2, 9, 10, 25
- Section 4.1: (pg 227) 2, 5, 11, 13, 19
- Section 4.2: (pg 234) 3, 6, 7, 9, 13
Homework 1
Due Friday, August 7 at 3:00 PM.
- Section 2.3: (pg 115) 5, 6, 9, 10, 12ab, 19, 21
- Section 2.5: (pg 132) 6, 8, 11, 32, 34, 35
- Section 5.1: (pg 269) 3ac, 7, 11, 14
- Section 5.2: (pg 282) 1d, 2c, 7, 8, 9, 17
- Section 5.3: (pg 288) 1, 4ad, 8, 10, 11, 12, 15
Solutions to 5.3.11 and 5.3.15, courtesy of Kim.
Homework 2
Due Friday, August 14 at 6:00 PM.
- Section 5.4: (pg 293) 3ac, 4ac, 7, 10, 14, 15
- Section 5.5: (pg 302) 5, 13, 17, 30
- Section 6.1: (pg 313) 3, 5, 9, 11
- Section 6.2: (pg 326) 1, 3, 7, 11, 13, 15, 24, 26
Note: The back-of-book answer for 6.2.3 is incorrect.
- Section 4.3: (pg 243) 4, 5, 8, 9, 12, 15, 16
Note: The back-of-book answer for 4.3.9 is incorrect.
- Section 7.1: (pg 356) 1, 4, 6, 7, 11, 12, 19, 27
Note: 7.1.1 is ambigious; add "travel counter-clockwise."
Note: 7.1.7 has a typo; it should say "from the point (-3, -3, -27) to (2, 2, 8)."
Note: For 7.1.19 you may (as always) use the integral table from the front/back of your text. If you want a challenge, this integral can be evaluated using techniques you know.
Note: The back-of-book answer for 7.1.27 is incorrect.
- Section 7.2: (pg 373) 3bc, 4, 11, 13, 17
Homework 3
Due Friday, August 21 at 3:00 PM.
- Section 7.3: (pg 381) 1, 2, 4, 5, 9, 13, 19
- Section 7.4: (pg 391) 1, 9, 10, 19, 21, 23, 24
- Section 7.5: (pg 398) 3, 5, 6, 7, 12, 15, 19
Note: The back-of-book answer for 7.5.7 is incorrect.
Note: For 7.5.19, "average value" is as defined in 7.5.18(a).
Note: The back-of-book answer for 7.5.19 is incorrect.
Homework 4
Due Monday, August 31 at 3:00 PM.
- Section 7.6: (pg 411) 1, 2, 3, 5, 6, 11, 15, 19
Note: The back-of-book answer for 7.6.3 is incorrect.
- Section 8.1: (pg 437) 6, 7, 9, 13, 15, 19, 20, 23
- Section 4.4: (pg 258) 15, 19, 34, 39
- Section 8.2: (pg 450) 8, 11, 13, 15 (hint below), 19, 31, 33
Note: I've typed up the rest of the example from the end of Wednesday's lecture, including an explanation of why the boundary is oriented that way, since we didn't have much time to explain that. Take a look: Stokes' Theorem Example
Note: For 8.2.15, note that the portion of the plane x+y+z=1 contained in the sphere is a disc centered at (1/3, 1/3, 1/3). Since the boundary of this disc is the same as the boundary of the surface S, applying Stoke's Theorem to each would lead to the same line integral (with appropriate orientations taken). Therefore, the surface integral of ∇ × F over S is the same as the surface integral of ∇ × F over this disc, which you may find much easier to evaluate.
In fact, you can do it without even having to parametrize that disc or find an antiderivative! Note that plane x+y+z=1 (and hence our disc) has a normal vector (1, 1, 1) which does depend on x, y, and z. Note that ∇ × F = (-2, -2, -2) also doesn't depend on x, y, and z (it also happens to be normal to the disc, but that is not critical for this trick). Then using Theorem 5 on page 406 of the text you will end up with the surface integral of a constant. This reduces the problem to simply finding the surface area of the disc using the geometric formula Area = πr^2.
Note: For 8.2.31(b), the circulation of F around C is just the line integral of F over C.
Homework 5
Due Friday, September 4 at 3:00 PM.
- Section 4.4: (pg 258) 3, 9, 36, 38
- Section 8.4: (pg 474) 5, 7, 11, 14, 15, 19, 29
Note: For 8.4.19, use the result of 4.4.38c); 'r' and 'r' are as defined there.
- Section 8.3: (pg 459) 2, 3, 6, 7, 10, 13, 16, 18
Note: For 8.3.18, there are typos in parts (a) and (d). For (a), there should not be a 'z' in the third component, which should just be 2xy+cos x. For (d), F(x, y) should be F(x, y, z).