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You can ask questions over our Review for Exam III by posting them here!
Archived May, 2026.
Questions on Review for Exam 3
User 001
Review Question 1)
When not given a time interval over which to evaluate a line integral, do we assume a standard interval of [0, 1]? Or is there an interval step I am missing?
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That's a reasonable thing to do, if I mess up and don't give you an interval, which I should have.
I'll update the problem but, yes, for the time being you can assume an interval of the form
0\leq t \leq 1.
User 002
Could you go over number 8 again? I am looking over the map projection notes but I am not able to find anything.
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Sure! This is actually very easy, once you understand a couple of ideas. First, though, #8 ask the following:
The equi-rectangular projection is the cylindrical projection defined by T(\varphi ,\theta )=(\theta ,\varphi ). Compute the general distortion factors M_{\varphi} and M_{\theta} for T as functions of \varphi and \theta. Use this to explain why the equi-rectangular projection is neither conformal nor equal area.
(Note that I mistakenly wrote M_m and M_p for the scaling factors along the meridians and parallels on the review sheet. I rewrote those and M_{\theta} and M_{\varphi} here. Both notations are common but I should be consistent.)
Now, the algebraic form of any cylindrical map projection is
T(\varphi, \theta) = (\theta, h(\varphi)).
The scaling factors along the parallels and meridians are
\begin{aligned}
M_{\varphi} &= \sec(\varphi) \: \text{ and } \\
M_{\theta} &= h'(\varphi).
\end{aligned}
Furthermore, the projection will be conformal if these are equal and equal area if they are reciprocal.
Next, the equirectangular projection has the formula
T(\varphi, \theta) = (\theta, \varphi).
That is the h(\varphi) function is the identity h(\varphi) = \varphi. Thus, h'(\varphi) = 1.
So... Putting these things together, the projection will be conformal if 1 = \sec(\varphi) (which it isn't).
And the projection will be equal area if 1 = \cos(\varphi) (which it also isn't).
This stuff is explained in the map analysis section of our class notes.
User 003
For question 2, I am thinking that the curve is in the same path as the vector field, so their dot product is positive and the line integral is positive as well.
However, the direction of the curve is confusing me, since it is negative, does that mean that the line integral is negative?
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