Prove that the work
produced along a oriented curve of depends only on the distances of starting and ending point of about the origin.
Let us consider the function
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.
Let be defined as
(i) Sketch the graph of .
(ii) Evaluate the line integrals:
(i) The graph of the curve is depicted below:
(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.
Let denote the dilogarithm function. Prove that
Take a loot at the JoM Forum
Let . Given the series
(i) Prove that converges forall .
(ii) Prove that is not a Fourier series of a Lebesgue integrable function.
Prove that there does not exist a sequence of complex polynomials such that uniformly on .
If such sequence existed then the convergence on the compact set
would be uniform.
which is an obscurity.