Homework #7 Questions
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.
.
for which 
.
.
.
for which 
.
for which 
. Choose a 
and 
and two different curves 
and 
such that the following three things hold:
.
? 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.
, the normal vector, 
, and the curvature vector, which combines the two, 
. 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.
.)
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 
(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, 
is measured in 
, and the projectile’s weight is given in 
, all the units in this problem work out without needing any conversions.
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 
(see Equation 11 and Example 5 in Section 14.3).
as unknowns in the definition of 
and 
) to do the check, so that’s 6 conditions. From these conditions you should be able to find 
and passing through 
, we get
.
, we get
.
.
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 
instead of 
, you are finding the plane made up of all the vectors perpendicular to 
point in different directions (they are always perpendicular to each other!), the planes you make from them will be different.
Math 317 Virtual Office Hours 