Q: Why can we need to define angle using v.u/(|v|*|u|) if we have already
algebraic definition of dot product? A: the formula v.w = |v| |w| cos(alpha)
links it to something geometric, the angle between the vectors.
The formula is a link between algebra an d geometry. We can compute the angles between two vectors.
For example, the angle between v=[1,1,0] and w=[0,1,1] is the arccos
of v.w/(|v| |w|)= 1/2 which is 60 degrees. The dot product alone does
not tell this. The dot product is in this case 1.
Q: Why is the dot product distributive?A: It follows from the distributive property we know from numbers. For example
3 ( [3,2,1] + [5,4,2] ) = 3 [3,2,1] + 3 [5,4,2] is in each component just
the distributive property you know from numbers. For example
3*(3+5) = 3*3 + 3*5 (for the first component).
Q: What is the intuition behind the proof of Cauchy-Schwarz?A: In the proof of Cauchy-Schwarz, one can think of v-a w as a
component perpendicular to the vector projection. Note that if |w|=1, then
the vector projection P_w(v) = (v.w)w/|w|^2 is equal to (a w),
where a=v.w is the constant defined in the proof. Now the Cauchy-Schwarz
inequality has a geometric meaning that the length of the projection is
smaller or equal than the length of the vector itself.e
Q: What about collaboration?
A: For homework, it is fine to discuss
the material with others, instructors but you have to write up things yourself. During
exams, absolutely no collaboration or consultation with others is allowed. This is strictly
enforced. Any anomalies would be investigated. And consequences would be severe. Look at the
Harvard honor code. There are reasons
why we want you to be seen in class, to submit homework and exams handwritten. These papers
will all be graded by people, not machines. If you plan to have somebody else do the work for
you, this is not acceptable. You don't want to be Biff from ``Back to the future"
Q: Do we need a textbook? A: you can take the course without
textbook. Our time is limited and it might be a good idea to focus on how we go through the
material which is very efficient. Textooks like Stewart have about 1000 pages.
A popular choice had been Multivariable Calculus: Concepts and Contexts (any edition).
I will say more about this in the first class.
Q:How much pre-knowledge is needed.
This is written in the description of the course.
Here is a 5 imes 5 checklist.
In general, multi-variable is a hard course. Also after teaching calculus for 30 years, there are
always new aspects to see, which makes it exciting. Especially geometry of three dimensional space
can be challenging but geometric thinking is important.
We will say something about it during the first lecture. If you are in doubt, write Oliver, as this
might depend a bit on circumstances.
Q:What about academic integrity?. We strictly follow to the
Harvard honor code. If you can not follow
this, you should not participate in any summer school course. Any violation of the code
is taken seriously and has severe consequences. Actual humans grade the assignments and we also
ask you only handwritten material and not typed up material which gives us some control to verify
that it was you did the work. If you have no confidence that you can follow
the code, you should not take this course (nor any other course at Harvard!).
Q:Do I have to participate actively? Yes,
the lectures are done by zoom. Having taught distance since a couple of years already in various
formats (first at the extension school in 2014 in a hybrid, and the first time using Zoom
in 2017)
and now in the spring of 2020
at the Harvard College, I have to ask you also to join the lectures with video.
This is the analog of showing up in person in a lecture.
There is the possibility to keep still some privacy
like using backgrounds in Zoom, in case you do not want to see other parts of your environment. We want you to
interact with teaching staff and peers like in person and we are all humans and not robots or avatars.
If you are uncomfortable with this, please choose a course, which allows ghosts to participate.
(If you are for some reason not able to join with video maybe by technical or other reason, please write
Oliver an email and explain. If the reason is legit, we can make an exception.)
Q:Will classes be recorded? A: This has been a frequent question already. The answer is no.
We will post some basic theoretical content as videos but the classroom will also feature
discussion and working on problems and to be a safe place, where one can make mistakes and ask any type
of questions without having this being recorded.
So, if you can not make it to the actual class, this is not the course for you. We understand
that this can produce difficulties in a different time zone. If it is impossible that you can
participate, chose an other course which meets at a different time. If we would meet all on campus,
you would also be expected to show up regularly to class. We understand that this can force you
a shift of time for a couple of weeks but we usually adapt quickly also when traveling.
Jet lag disappears in a few days.
Q:How much time you have to spend per week?A: There are 6 hours of lecture every week,
4 homework units per week (each expected to take each 2 hours in average), discussions with instructors
in office hours or problem sessions (about 2-4 hours in average) and then you need time to digest,
read, polishing notes etc. In average, students spend about 15-20 hours of uninterrupted time
per week for such a course. This is
an average, some use more time and of course as more time you think about math as better.
So, if you can not allocate this time, this course is not for you. Mathematics is hard to learn in
general, like anything which is precious.
Q:How are assignments submitted?A: Homework and exams are submitted electronically in PDF format. We ask you to hand-write and not
to type you work up. We can read between the lines from actual handwriting also have more
possibility to check academic integrity. It is important to work on paper (or some
electronic device like an Ipad, where you can write like with paper and pencil. This counts still as
handwritten, as long as no text mode is used). If you submit work on paper,
use a scanning app on the phone which produces good PDF documents from work done on actual paper.
Each unit is submitted separately as
it will be graded by different people in the teaching team. You should be fine if you have a scanner or
an app on your phone which can scan documents which consist of one PDF document. Note that each
unit is submitted separately. We will make some recommendations in the first lecture.
Q: Is this course curved?A: There are different things which students
usually understand with ``curving", some associate with it fixed cut-off levels,
others quote like a certain percentage for each grade bin. Neither is implemented here
(nor is it in most college courses). We are forced at the end to assign a letter grade to
numerical values and this by nature defines a curve. It is a function f(x) = y, where
y is the letter A-F and x is a number 0-100. The function f will be determined at the end.
By experience, the x-data distribution usually is a tilted Gaussian distribution.
Every student will at the end know both the value x and y.