The following problems will guide you to gain deeper understanding of the
key concepts in this course.
Please attempt as many problem as possible.
You can earn up to **20 points** for each problem.
You must type your solutions and clearly label each problem
(ideally each problem should on a separate sheet of paper).

Physicists and computer programmers all use the word “vector”, but they mean different things. Explain what does the word mean in these two contexts. How are they related to the mathematical meaning of that word?

How is the length (magnitude) of a vector is defined? Why is it defined that way?

We have demonstrated that matrix-vector product using 2x2 matrices represent linear transformations of the plane. Write down matrices that represent the following transformations:

- Counter clockwise rotation by an angle
- Clockwise rotation by an angle
- Uniform scaling by a factor of 2
- Scaling by a factor of 2 in the direction only

How can we represent rotations in 3-dimensional space using 3x3 matrices?

For a 2x2 matrix

\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]

its determinant is . The book told us that this is the “signed area” of the parallelogram spanned by the two column vectors. Explain why.

The magnitude of the cross product between two vectors is the area of the parallelogram spanned by the two vectors. Explain why.

The brachistochrone problem is a famous mathematical problem posted by Johann Bernoulli in 1696 to all the best mathematicians in western Europe. This simple yet fascinating problem sparked the development of several branches of modern mathematics (arguably including calculus of variation). Provide a brief description of the problem, its history, and its solution(s).

For a function in two variables, that is a gradient vector of ? From the point of view of geometry, does it mean anything?

For a function in two variables,
what is the definition of a *directional derivative* of this function?
How is it connected to partial derivatives?
Are partial derivatives special kinds of directional derivative?

A 4-dimensional sphere is the set of points in whose distance from the origin is 1 or less. Using multiple integrals, can you compute the volume of such a sphere?

As a natural generalizations of the previous part, can you compute the volume of an -dimensional unit sphere?

Plot the volume of an -dimensional unit sphere as function of the dimension and observe the general trend. What do you notice? Do you have an intuitive explanation of this trend? What would be the limit be as ?

Can you visualize a 4-dimensional sphere? How?

To make the computation easy, let us assume the earth is a perfectly spherical
ball with a uniform density same as that of granite.
What is the gravitational pull experienced by a point mass of 1 kg
that is 1000 meters from the *surface* of the earth?

Let’s suppose there is planet that has a hollow core (whether or not this is physically possible is a totally different question). For simplicity let us assume this planet is actually a perfect spherical shell (no mountains of valleys) of certain thickness and uniform density. What kind of gravitational pull will a person inside the hollow planet experience?

A *planimeter* is a mechanical measuring device that could accurately compute
the areas of *arbitrary* shapes.
In certain sense, it is a mechanical manifestation of the famous Green’s theorem.
Explain how such a device could work (using Green’s theorem).

This YouTube video, shows a planimeter made from a spoon. Does it really work? Why?

Before the invention (or adoption?) of calculus, planetary motion
(how a planet would orbit the sun) was described using *Kepler’s laws*.
What are these laws?
How can they be explained using multivariable calculus?