## 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

### That crazy planet you’re on

Over thanksgiving we have a homework problem about waking up on a strange planet.  One student has asked:

Q.  Regarding the second hw problem–estimating mass of planet–do we have infinite/sufficient amount of time?  And could I assume the ruler has infinite length, and stopwatch can time infinite length of time period? (since they’re both of “very high quality”)

It’s an open ended, creative, problem.  You can answer these questions however seems reasonable to you.  It might be going kind of far to say the ruler is infinite, since then one might wonder about the question of whether or how an infinite ruler could have fit into your pocket or backpack, or whether such a thing could exist in our universe.  On the other hand, maybe those sorts of questions don’t bother you.  You could assume the ruler is as long as you like to start with, and then, figure out how long you actually need it to be, and ponder the question of how to make a long ruler using a shorter ruler.  Have fun with it, and write up what you think is the interesting and relevant part of your ideas.  I’ll grade the problem based on how well thought out and mathematically sound your response is, not how much it resembles some particular idea I have in mind.

### Pre-Midterm Questions

Q.  I was wondering what curvature vector really is and why it is important.

We studied curvature, $\kappa$, the normal vector, $\mathbf{N}$, and the curvature vector, which combines the two, $\kappa\mathbf{N}$.  All three things tell us about the curviness of a curve.  If curvature is large, the curve is more tightly bending or curving.  But to tell you that a curve is bending tightly is not complete information:  you would also ask me which way is it bending.  Imagine a car driving around a tight corner.  You might ask me, “is it turning left or right”?  In the plane, the normal vector points either to the left or right from the point of view of the particle travelling along the curve.  In space, there’s more freedom of direction.  If I tell you a spaceship is making a sharp turn, you might ask, “is it turning up, down, left or right or somewhere in between those?”  The normal vector tells us which direction the turn is in.  Finally, we sometimes use the curvature vector to combine the two pieces of information into one mathematical quantity, which is useful since they are so closely related.
Why is it important?  Simply because it turns a basic concept applicable to curves, the “bendiness” or “curviness” into a mathematical quantity which we can then study with calculus.  Calculus is a language for physics, historically, and a language for a much wider range of phenomena in life now.  So each concept of the physical universe, such as “bendiness” needs to be translated into calculus, so that we can use it precisely and quantitatively to build bridges, test new drugs, and study far away stars. The dorky examples in your textbook may not be convincing, but that’s only because using calculus in real life is usually complicated by so many details that it ceases to have pedagogical value.

Q.  I saw that the sample midterm doesn’t have the formula for the two components of acceleration.  In general, do we need to memorize any formula for this midterm?
I have said that I don’t like memorizing formulas, and I don’t:  one just forgets them anyway, and they are very easily accessible in any real world situation (you rarely have to do calculus in the jungle without your iphone).  However, a midterm is a contrived testing situation.  So yes, you should know your formulas.  You can depend on the ones on the front page of the sample midterm being on the real midterm.  But any other formula you find yourself using during homework and review you should either a) store in your noggin; or b) know how to re-derive it if necessary.  (I’m of the “re-derive” it school of thought, but some take too much time to re-derive, such as the more complicated formulas for curvature, in a testing situation.  The components of acceleration, should you forget them, can be re-derived without too much trouble: use product rule to differentiate $\mathbf{v}' = v\mathbf{T}$.)

### Virtual Office Hour on Hwk #4

Q. (14.4 #30) I’m not sure if 4N is a parameter in physics that has its own meaning or it’s the normal vector or it’s just simply a letter.

Thanks for asking, as others might be wondering about this too.  The letter $N$ stands for “Newton”, which is a unit of force.  So this is wind blowing with 4 units of force.  A Newton is the same as one $kg\cdot m/s^2$ (a kilogram metre per second squared).  Remember that force is mass times acceleration and this makes some sense:  a kilogram is mass, and metres per second squared is a unit of acceleration.  Since the gravitational acceleration of the earth, $g$ is measured in $m/s^2$, and the projectile’s weight is given in $kg$, all the units in this problem work out without needing any conversions.

Q. (14.3 #60) I’m not sure how to use the knowledge we learned to get such a polynomial. I can guess by using graphing program but I can not associate it with the things when learned in class.

This question asks you to find a polynomial of the form $P(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f$

so that the piecewise function $F(x)$ in the problem is a) continuous; b) has continuous slope; and c) has continuous curvature.  The relation to what we’ve done in class is mostly in part (c).  We learned how to find the curvature of a function like $P(x)$ (see Equation 11 and Example 5 in Section 14.3).

So my suggestion on how to start:  leave $a, b, c, d, e, f$ as unknowns in the definition of $P(x)$ and $F(x)$, find the slope and curvature of $F(x)$, and then check whether a) $F(x)$ is continuous, b) its slope is continuous, and c) its curvature are continuous.  (By check whether they are, I mean figure out what needs to be true about $a, b, c, d, e, f$ in order that they are.)  There are two important places ( $x=0$ and $x=1$) to do the check, so that’s 6 conditions.  From these conditions you should be able to find $a, b, c, d, e, f$ that will make all the necessary things continuous.

Q.  I was wondering if we could use simple T, N, and B vectors for simplicity when finding equations for normal and osculating planes. I did this method for 14.3 #46.  I just found those vectors without making them as unit vectors (because unit vectors get complicated).  I think this is still correct since the ratios between x,y,z are still the same without being unit vectors.  Am I correct? And if so, can we just simplify thru this method for quizzes and midterm?

Yes, you are correct.  Suppose we have two vectors in the same direction, $\mathbf{v}_1 = \langle 1, 2, 1 \rangle$

and $\mathbf{v}_2 = \langle \sqrt{2}, 2\sqrt{2}, \sqrt{2} \rangle$

If we form the equation of the plane with the normal vector $\mathbf{v}_1$ and passing through $(0,0,0)$, we get $x + 2 y + z = 0$.

But if we use $\mathbf{v}_2$, we get $\sqrt{2}x + 2\sqrt{2}y + \sqrt{2}z = 0$.

The second equation can be written $\sqrt{2}(x + 2y + z )= 0$.

And multiplying the entire equation by $\sqrt{2}$ does not change the solutions, or the plane it represents.  So these two planes are exactly the same!  So yes, you can change the normal vector by any scalar multiple in order to make it more convenient to write the plane.  In particular, you can just find $\mathbf{T}'$ instead of $\mathbf{N}$, because these two vectors are in the same direction.

Q.  I was curious about #46 of 14.3. In the textbook they calculate the normal plane using the vector T(t). Is it equivalent to use the vector N(t) to calculate the normal plane? The equations for the planes in each case are different but are they inherently the same?

No, they are different.  When you calculate a plane from a vector $\mathbf{v}$, you are finding the plane made up of all the vectors perpendicular to $\mathbf{v}$.  Since $\mathbf{N}$ and $\mathbf{T}$ point in different directions (they are always perpendicular to each other!), the planes you make from them will be different.

Part of the confusion here might be the names.  The word “normal” in calculus usually means “perpendicular.”  But of course, this begs the question, “perpendicular to what?”  The “normal vector” $\mathbf{N}$ is perpendicular to $\mathbf{T}$.  The “normal vector” of a plane  (the vector you make the plane from) is perpendicular to the plane.  So these are two different sorts of “normal vectors” and they have nothing to do with each other!  Very frustrating, but we’re stuck with it for now.