Frequently asked questions
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Question:
Q: What is the curve?
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Answer:
You see the curve to the left. There are about three constraints:
there should be around 40 percent A and A-, the average should be in the B+ range
and the standard deviation window should include only A and B grades. Then,
according to the distribution, the cutoffs are set. This also depends on gaps in the
distribution, i.e. avoiding too flat parts of the curve for a cut.
The cuts also depend at the analysis of grades of a few "gauge" students near
the threshold.
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Question:
Q: Do you have statistics about resurrection?
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Answer: Sure. Resurrection usually takes place for about 10 percent of students.
Because the HW score were in general high, beating the average grade with the final grade
is not so easy. But resurrection happend, for example, with
hard work. The tougher material of the third part could make
a difference. The graphics to the left shows the correlation between
average score and final score. The graphics to the right shows the
difference of the final score minus the average score.
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Question:
Q: What does
rnyyl? Nyy qbar? Erzrzore gung ceboyref ner rnfl vs lbh xabj gur nafjref
on the exams page mean?
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Answer:
You have to find out yourself. Hint: it is is a
Cesar cypher
The one used is quite special and has its own name.
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Question:
Q:
In many of the problems I encountered, it
mentioned to assume the region is a simple solid. I understand that this is
because the theorems only work under these conditions. However, the book
only gives two examples of simple solids. It says a simple solid is
something like a rectangular box or an ellipsoid, but it never says how to
determine if something is a simple solid-it never mentions what something
that isn't a simple solid is. What determines if a solid is "simple"?
What happens when solids aren't simple? How would one go about calculating
things like flux?
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Answer:
Stewart calls a region simple, if it is
both Type I, Type II (p 934). In three dimensions,
it should be os that you can set up a triple integral
with any given variable order and the integrals of
the third bound are given by functions (p.966).
The definition of "simple solid curve" not so important.
Dont mix it up with "simply connected". That refers
to the property that you can pull together any
closed loop in the region to a point.
For the integral theorems, the notion of simple
does not matter. It is just simpler to setup both
the triple integral as well as the surface integrals
if you have a simple region. But it can also be no
problem for other regions like the doughnut, which is
not simple but still, it is still (relatively) simple to
parametrize the boundary.
If you should have a solid, for which you have several
surfaces as a boundary, you will have to parametrize all
the boundaries. An example is a swiss cheese. You have to
parametrize the outside, as well as the holes. A swiss
cheese is not a simple region.
The notion of simply connected only enters at one place
in our course. If a region is simply connected and a
vector field has curl(F)=0 in that region, then there
is a function f such that F = grad(f).
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Question:
Q:
Will we be expected to derive any results e.g. how to get from rectangular to polar
coordinates in a double integral. Or will we be expected to just apply these
results?
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Answer:
In general, you don't know to know the proofs. But note that
in order to remember the r factor for example, when
going to polar coordinates, it is good to have in mind
the little wedge of size dr and r dtheta .
The area of this gives you the distortion factor r dr dtheta
While you have not to know the derivations of such things
in an exam, it will help you to remember this in the long
term. Similar with other topics of this midterm: to remember
the Lagrange equations, it is good to have in mind that
these equations just express that two gradients have to
be perpendicular and if the two gradients were not
perpendicular, then the rate of change of the function
along the constraings is nonzero.
The second derivative theorem is a result which is probably
best remembered for now without proof.. One needs linear algebra
to give an intuitive proof. The proof you find in the book is a
hack, not intuitive.
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Question:
Q: In chapter 9 about curves, there are a lot of formulas. Which do we have to know?
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Answer:
- Velocity, speed, acceleration, the unit tangent vector, how to get r back from r',
arc length have to be known
- Curvature formula, the formulas for the Frenet frame have not to be known by heart,
but you have to know what curvature means.
- You will not have to memorize the curvature formula. But when given, you can compute
the curvature like here in the case of the helix:
r[t_]:={Cos[t],Sin[t],t};
v=Cross[ r'[t],r''[t]]/Sqrt[r'[t].r'[t]]^3; Simplify[Sqrt[v.v]]
where the curvature is constatn.
- Here is a proof of the curvature formula with mathematica
r[t_]:={x[t],y[t],z[t]};
s[t_]:=Sqrt[r'[t].r'[t]];
T[t_]:=r'[t]/s[t];
kappa1[t_]:=Sqrt[T'[t].T'[t]]/s[t]
v[t_]:=Cross[r'[t],r''[t]];
kappa2[t_]:=Sqrt[v[t].v[t]]/s[t]^3;
kappa1[t]^2==kappa2[t]^2
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Question:
Q: Can I use the old edition of Stewards book?
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Answer:
We need the third edition. The problems are different from
edition to edition.
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