## Q.  On what questions can we use the Fundamental Theorem of Line Integrals or knowledge from section 17.3?

There’s some confusion about whether we’re allowed to use the Fundamental Theorem of Line Integrals for homework.  You are allowed to use it only for the textbook questions from 17.3.  It’s such a great, useful theorem, that you’ll now be tempted to use it for questions in 17.2, but for now, you’ll learn more by doing it “the hard way”.  In particular, Extra Problem #3 and the Bonus Problem should be done without it (see the two posts immediately before this one for these problems).

## Q.  For #43, can we use the fact that the force of gravity function is conservative?  [For the bonus?]

You could conceivably get an answer to #43 in at least 3 fundamentally different ways:

1) You can use a parametrisation of the helix;

2) You can directly argue why the parametrisation of the helix doesn’t matter in this case; and

3) you can use conservation of the graviational force and the big fundamental theorem of line integrals.

You learn the most by doing it all three ways, but my intention for this assignment was:  full marks for (1); bonus marks for (2); and not allowed to do (3).  The solutions manual does (1), and I suggest you do this first.  Then, you’ll see that the calculation simplifies in certain ways.  What I am looking for as (2) is to explain in a down-to-earth way (nothing about gradients or conservation) why one expects the simplification in the calculation and can therefore exploit it to avoid using the parametrisation of the helix at all.

### Extra Problem on this Homework #7

Hi all,

I’d like to add a problem to the homework assignment, but seeing as it is Sunday, I don’t feel it’s fair to add it now.  So I’m going to add it as a bonus problem.  It isn’t a bonus because it’s hard; it’s just a bonus because I only thought to add it now.

For a 2-point bonus problem:

Pertaining to question #43 & #44, explain how and why you can do the problem without using any parametrisation of the helix.  Feel free to do it first using a parametrisation, but then examine your solution and then re-do it without any parametrisation.  A well-explained reasoning for why it can be done this “short-cut” way will receive full marks.  You can label this a separate problem, or you can just work it into your #43/44 solutions.

Kate.

### Homework #7 Extra Problems

What follows are extra problems for Homework #7.

## Bonus Problem.

I added a small extra problem here.

## Problem 1.

Take a look at this vector field $\mathbf{F}$.

For each part of this problem, please provide a justification (it need not be a calculation:  it can be an intuitive explanation).

a) Identify this vector field (i.e. give a likely formula for $\mathbf{F}$).

b) Give an equation for a closed (i.e. closed loop), non-trivial (i.e. not just a point) curve $C$ for which $\int_C \mathbf{F} \cdot d\mathbf{r} = 0$.

c) Give an equation for a curve $C$ for which $\int_C \mathbf{F} \cdot d\mathbf{r} > 0$.

d) Give an equation for a curve $C$ for which $\int_C \mathbf{F} \cdot d\mathbf{r} < 0$.

## Problem 2.

Take a look at this vector field $\mathbf{F}$.  I don’t expect you to be able to figure out what vector field it is (i.e. what equation gives it):

Either print out this vector field or draw a rough copy of it in your notes (showing the main features but with far fewer vectors; don’t waste too much time).

a) On this vector field, draw a closed non-trivial loop $C_1$ for which $\int_{C_1} \mathbf{F} \cdot d\mathbf{r} > 0$.

b) On this vector field, draw a closed non-trivial loop $C_2$ for which $\int_{C_2} \mathbf{F} \cdot d\mathbf{r} < 0$.  Choose a $C_2$ which does not intersect $C_1$.

c) On this vector field, draw two different points $P$ and $Q$ and two different curves $C_3$ and $C_4$ such that the following three things hold:

1) $C_3$ is a curve with endpoints $P$ and $Q$ and orientation going from $P$ to $Q$;

2) $C_4$ is a curve with endpoints $P$ and $Q$ and orientation going from $P$ to $Q$;

3) $\int_{C_3} \mathbf{F} \cdot d\mathbf{r} < \int_{C_4} \mathbf{F} \cdot d\mathbf{r}$.

## Problem 3.

Let $\mathbf{F}$ be a constant vector field in the plane (2D vector field).  Is there a closed loop $C$ in this field for which $\int_C \mathbf{F} \cdot d\mathbf{r} \neq 0$?  Why or why not?  (If there is, write it down.  If there is not, prove there is not.)  NOTE:  If the answer is a proof, then I would like a direct proof, not one based on the Fundamental Theorem of Line Integrals or anything from section 17.3.  This is a simple example where you can check it out directly.

### News from This Week

Attendance is low this week and many exams haven’t been picked up.  If you’re missing class, please view the page “Where we are” on this blog to see what you have to catch up on to be up to date with this class.  We are well into Chapter 17 now.  There was a handout/worksheet on Monday which is up on the webpage for download. (Click on “Online Demos” and then look for “Resources by Lecture” and look under October 13.  Try to complete this worksheet yourself, using the textbook as a guide, and then compare to answers which will be posted later).

In response to student requests (and an overwhelming vote) I am allowing students who do 15% better on the second midterm to count that midterm grade for their first midterm as well.

Also, we now have a “party funds tally” page on this blog, where I’ll keep a running tally of our party funds.  You can leave comments on that page about what we should do for the party.

Kate.

### Midterm #1 Solutions

Midterm Solutions for Test A & Test B (look at the header of your test to figure out which is the right one for you).  You’ll also find these listed on the main website under “homework”.

### Midterm #1 Histogram

You can view it here.

### Hwk #5 Problem 1

I hope you all had the kind of long weekend you were hoping for: relaxing, frenetic, productive, or replete with bird.

I’m just giving a last minute hint on Problem 1, in case you’ve gotten stuck.  First, in a long narrow ellipse, where are the foci?  And second, if the ellipse gets really really really narrow, you get a line segment — where do the foci go then?  In calculus, we can answer questions by answering them approximately, then taking the limit.  Here, the ellipse is getting narrower.  The trick is all in drawing the right picture: what is the major axis of the narrow ellipse?

For the other problem, I’m not giving out hints, because it’s an open ended problem, and up for interpretation.  Some people have asked, “are you allowed to do X?” and I’d say, pick the answer to that which gives the most interesting answer to the puzzle.  Have fun with it.

Kate