The line is the shortest path between two points

Let be two points of , be a unit vector and let be a curve passing through those points. (that is there exist such that .) Prove that the shortest path between these two points is the line.

Solution

The following facts hold:

(1)

(2)

Integrating we get that:

(3)

where is the length of the curve between the points . Taking we have

Order of a group’s element

Let be a group and such that and

(1)

where is the identity element of the group. Find the order of .

Solution

We will begin stating a lemma:

Lemma: If then .

Proof:

First we multiply with and from right and left respectively. Thus one can see that

(2)

Thus

(3)

and

(4)

Now, we use the main relation and so

By repeating the previous procedure, one can prove the result.

Using the lemma we see that and thus . Since is prime the order of will be either or .

A contour integral

Define

Evaluate the contour integral .

Solution

We are applying the substitution thus:

since the function has only one pole in the specific contour , namely .

Finite matrix group

Let be a finite subgroup of   this is the group of the invertible matrices over ). If then prove that .

Solution

Let us suppose that and . We note that for every the depiction such that is and onto. Thus:

Thus the matrix is idempotent. thus its trace equals to its class. (since we are over which is a field of zero characteristic.) Hence

This implies that hence .

The exercise can also be found at mathematica.gr

The volumes are equal

Prove that for every constant the set

has the same volume for all continuous functions .

Solution

For every on the plane the set

is a disk of constant radius . Thus the set

is a “cylinder” which axis is the curve

and its radius is .More specifically , the set is bounded by the planes and and for every the intersection of with the plane is the disk

The area of this disk is the same with the disk

The latter one has an area of

It follows from Cavalieri’s Principal that has the same volume and that is equal to

which is the same for all continuous functions .

A somewhat visualization would be the following:

This was an exam’s question somewhere in Greece. The answer was migrated from mathematica.gr .