## Work along an oriented field

Prove that the work

produced along a oriented curve of depends only on the distances of starting and ending point of about the origin.

Solution

Let us consider the function such that

which is continuously differentiable in and it holds that:

Hence the vector field is a conservative one. This , in return,  means that is actually independent of the road we choose , meaning that it only depends on distances of starting and ending point of about the origin.

This was an exam’s question somewhere in Greece. The answer to this question was migrated from the Greek team of mathimatikoi.org forum.

## Curves and line integrals

Let be defined as

(i) Sketch the graph of .

(ii) Evaluate the line integrals:

Solution

(i) The graph of the curve   is depicted below:

(ii)

(i) For the first integral we have successively:

(ii) For the second integral we have successively:

This exercise was an exam’s question somewhere in Greece.

## A dilogarithm and log Γ integral

Let denote the dilogarithm function. Prove that

Solution

Take a loot at the JoM Forum.

## Not Lebesgue integrable function

Let . Given the series

(1)

(i) Prove that converges forall .

(ii) Prove that is not a Fourier series of a Lebesgue integrable function.

(i) Let .We note that for the series , trivially, converges since . For all other we are using Dirichlet’s test. It is well known that is bounded and we also note that . The result follows.

(ii) Suppose that such exists. Integrating (we can do that since Fourier series is integrated term by term) we take a continuous and of bounded variance function. The Fourier series of this function should converge at . This is not the case here since is known to diverge. Thus, such function does not exist.

## Not a uniformly convergent polynomial sequence

Prove that there does not exist a sequence of complex polynomials such that uniformly on .

Solution

If such sequence existed then the convergence on the compact set would be uniform.

However,

which is an obscurity.