Question:
When do we write int F(r(u,v)) . r_{u} x r_{v} du dv
and when do wr write int F(r(u,v)) . n |r_{u} x r_{v} du dv
for the flux integral?
Answer:
The two expressions are the same. Our book uses the second notation with the unit normal vector n. There
are several reasons, why it is better to avoid it:
it is a more complicated notation
it seduces you to compute the unit vector n which is not necessary.
It does not make sense if the surface has places, where the vector r_{u} x r_{v}
is zero. The first notation still works.
Question:
How do you define parallel vectors? Do they have to run in the same direction?
or can they run in opposite directions and still be parallel?
Answer:
Parallel vectors can also point in opposite directions.
The vector (2,3,4) is parallel to (-8,-12,-16).
The zero vector is parallel to any vector. Two vectors
are parallel if and only if their cross product vanishes.
Question:
I am trying to do Episode 4 practice exam, and I'm confused about problem 5:
how do you get f(x,y)=x^4+10x^2+25+y^4-2y^2+1? I can see that
F(x,y)= is a gradient field, but I don't get where the f(x,y)
comes from.
Answer:
you have to extremize the length of the vector field, not the potential
To extremize the length, you can also
extremize the square of the length.
While you are right that the vector field is a gradient field,
this is irrelevant in this question. There are different
ways how you can derive a scalar field from a vector field:
F -> f
if F = grad(f)
F -> curl(F)
in two dimensions
F -> div(F)
in any dimensions
F -> |F|
in any dimensions
In all these cases, the middle entry is a scalar field. In the problem
of your question, we look at the length of the vector field.
Question:
In the Fridays review, in the flowchart shown, one of the boxes said to
check to see if a given vector field is the curl of something else. Is there
any quick method to checking that?
Answer:
yes, there is. If the divergence is zero everywhere in space then
there the vector field is the curl of something else. There are
some cool integral formulas which give you G such that F=curl(G) but they
were not covered in our course. We do not assume that you know this and
we would give you the vector field G. There are cases, where one can
guess it like F = (0,0,2), where one could take G = (-y,x,0).
Question:
Could you clarify for what parts of the homework Mathematica usage is
allowed (e.g. when should we use Mathematica or not)?
Answer:You can use Mathematica for all your homework. But you
have to include all steps of your computations. For example,
if you use Mathematica for checking the result of an integral, you
still have to include all integration steps. For example,
if you compute a triple integral, we need to see how you compute the
most inner integral, then the middle integral and finally the outer
integral. If you draw something
with Mathematica, we need to see your code. Just printing out the
notebook you created is ok. It is often much easier to plot it
by hand. In general, you have to acknowledge all use of
Mathematica and document, what you do.
Question:
Why do you test TF questions in exams. They are often hard.
Yes, we know. They are tricky but they are also fun.
We want you to learn the material as efficiently as
possible. There is a definition aspect (know the terms)
an algorithmic aspect (compute stuff) and a conceptional
aspect (understanding things) in learning. Experience
shows that asking too many conceptional problems as major
problems in exams can be frustrating, both for students as well as the
graders. Multiple choice questions are an elegant way to
include this level nevertheless. TF questions push the boundary
of your knowledge, they can allow you to detect and fix
misconceptions, which you would not see when practicing
algorithmic tasks only. We always advise not to spend too much
time with the TF problems during the exam,
but to think about them well when doing the practice exams.
Question: I have trouble understanding section 13.3.
Don't worry, you can skip this. Epsilon delta definitions of continuity are not very intuitive. It usually
is hammered into the brains by intimidation and repetition. The mathematician Edward Nelson gave once
the following example: compare the usual definition of continuity: a function f is continuous at x_{0} if
> 0 > 0, so that if |x-x_{0}|< then |f(x)-f(x_{0})| <
with the following statement:
> 0 > 0, so that if |x-x_{0}|< then |f(x)-f(x_{0})| <
Euler
What does the later statement mean? --- You probably had to think hard for a while to get it. Everybody
has! Try it out with your friends taking math 23a. This proves that the Weierstrass notion of continuity is
unintuitive. And as usual with didactics, only experiment with real people reveals this.
An epsilon-delta definition is a good working definition, if you do formal proofs, if you are a mathematician.
Euler has a much better definition: f is continuous at x_{0}, if for all x close to x_{0} the value f(x) is
close to f(x_{0}). It is as precise as the Weierstrass definition, if one defines "close" appropriatly.
This is done in nonstandard analysis, which clears up the most dreadful parts of calculus. It never made it to the
calculus curricula because it would make half of the calculus teachers oblivious ...
Weierstrass
By the way, do you know the shortest joke mathematicians tell each other? Here it is:
= 0
Don't think, it is funny? Well, it only works on mathematicians.
Question:
Can you explain the notations exp and log?
Answer:
The function exp is the exponential function. It means exp(t) = e^{t}.
The function log is the logarithm function. Unlike in high schools, where one often
uses ln(x) for the natural logarithm, the logarithm is so important that it is called
log in most of the mathematical and scientific literature. The logarithm to the base 10
is not natural. It happend just by accident that humans have 10 fingers. Also Mathematica
uses Log for the natural logaritm.
The function log is the inverse of exp: one has
exp(log(x)) = x
log(exp(x)) = x
Note that d/dx log(x) = 1/x for positive x and d/dx exp(x) = exp(x).
Question:
I don't understand what the ways of converting back and forth from parametric
and implicit are. "Read off the radius" or "know it" make no sense
to me. Also, maybe you could conceptually explain the ways that parametric and
implicit are converted back and forth for the other surfaces as well.
Answer:
There are not many surfaces, where you can switch forth
and back from implicit to parametric. In general, it is
not possible. These are two different worlds and going
from one to the other is not easy. There are no general
tools to do that. What we can do are 1) Spheres and general quadrics
2) surfaces of revolution 3) planes, 4) graphs.
I don't know for example, how to parametrize general
Kummer surface.
One could solve for one of the variables but then one
would only parametrize part of the surface which are graphs.
So, in general we are completely helpless in finding
parametrizations of implicit surfaces and finding
an implicit description of parametric surfaces.
Question: Do we need a graphing calculator for the
first HW problem in the 4th week?
Answer:
Absolutely not. This was just mentioned if you want to have som fun
with a toy. You have just to write down an equation of a surface in cylindrical
coordinates and an equation of a surface in spherical
coordinates. No graphing calculator is needed for that.
Now, you have to be able to draw this surface.
For example, your homework solution to the two problems
could look as follows, if it were not too simple.
PROBLEM A. My equation is r = 1. This is a cylinder. Here is
a picture.
PROBLEM B. My equation is rho =1. This is a sphere. Here is a
picture.
Start with simple things like r = z (a cone) or
rho = 1/cos(phi) which is the plane z = 1. If you use too
complicated formulas, you will no more be able to analyze or draw
the surfaces.
Question:
Can the radius r in polar coordinates become negative?
Answer: many calculus books including our book allow that when drawing
polar curves like r = sin(theta). In general, it is better to avoid
negative radii. The radius is the distance to the origin. If one allows
negative r, then one adds more ambiguity into the point (-1,0) could be
written as (r,theta) = (1,Pi) or (-1,0).
Question:
I have some problems with problem 32 for Monday October 16.
Answer:
It is best to look at the normal vector n = (-2t,1)
to the curve and not the tangent vector (as the hint says).
The reflected vector of the vector v = (0,-1) is with the
projection vector P_{n}(v) given by
w = v-2 P_{n}(v)
Now the computation is easier:
r(t) = ( t , t^2 ) point on parabola
r'(t) = ( 1 , 2t ) tangent vector
n(t) = ( -2t, 1 ) normal vector
v = ( 0, -1 ) incoming vector
w = v-2 (v. n ) n/(1+4t^{2})
= ( 0, -1) +2 ( -2t/(1+4t^{2}),1/(1+4t^{2}) )
= (-4t/(1+4t^{2}),-1+2/(1+4t^{2})) reflected vector
reflected ray parametrization
(t,t^{2}) + s ( -4t/(1+4t^{2}),-1+2/(1+4t^{2}) )
= (t - 4 t s/(1+4t^{2}), t^{2}-s+2s/(1+4t^{2}) )
Look for the s for which the first coordinate is zero
and show that then the y coordinate is constant independent of t.
Question:
Do we have to know section 12.5 in the book?
No.
Question:
We have covered curvature only for a very short time in class. What do we have
to know about curvature in the exam?
While the topic of curvature is exciting, we only scratched the surface. You do
not have to know curvature formulas nor know about osculating circles. We recommend
however that you read about it. It is a nice topic. By the way, in order to compute
curvature like in the homework problem, it can pay off to plug in numbers early.
If you want to compute the curvature at a specific point, it is enough to compute the
velocity vector r' and acceleration vector r'' at the point and then compute the curvature
with |r' x r''|/ |r'|^{3}after you have plugged in the time t.
If you carry along the full computation with the variable t until the end, you might
get into a terrible mess.
Question:
Q; On problem 42 in section 12.3 of the Math 21a homework, the
the arc length function is far too complex. Mathematica gives
a nasty solution.
Humans are still better than machines!
The core problem is that Mathematica does not simplify Sqrt[x^2] to x.
Mathematica produces from
which is of course equal to (3/4) cos(2t) but Mathematica
does not simplify (because taking square roots is a source of errors).
A human can simplify because sigma(t) is nonnegative
Question:
Q; I am looking at section 12.3 and am confused about some symbols the book is
using. On page 80, under the definition, it says u |--> r(t0) + ur'(t0). What
does the arrow |--> mean? I've never seen that notation, and it comes up in #50 of
our problem set. Is it just another way of defining a curve parameterized by a
vector? If so, would that mean that in #50, it's just talking about two curves
C1(t)= and C2(t)=<3t+1, 2t, t^2+t-1>?
Instead of f(x) = sin(x), one also uses the notation x |--> sin(x)
This allows the autors of the book to avoid introducing a new function on
page 80. The tangent line in this case is again a parametrized curve, one
can write it for example as
R(u) = r(t0) + u r'(t0)
For the tangent curve, one has taken a new parameter u. It is an other parameter
than t.
Question:
Q; There are some problems in the book which are labeled "Computer exercices"
We rarely assign those, but if you you have one, why not get started
already with Mathematica and do it with Mathematica. You should also
be able to do it with a calculator, but we strongly suggest to get
away from calculators and use a more powerful computer algebra system.
Question:
Q; The coop is out of text books.
Sorry about the invonvenience.
We might post the problems to first week online and investigate
why there are not enough copies. It is sold on online shops
like Amazon but we don't know how long it takes to ship
at the moment.
Question:
Q: Why do we use an other book this semester?
Answer:
There are several reasons to try an alternative to Stewart: Solutions for Stewarts book
are now widely available online. This happend especially after CDs of the solutions
were available and many students had access to solution books.
The prize of Stewart is unreasonable. The new book of Blanck and Krantz contains all we
need and is half the prize.