## Is it a conservative field?

(i) Let be the unit disk and be its positive oriented boundary. Evaluate the line integral

(ii) Can you deduce if the function

is a conservative field using the above question?

**Solution**

(i) We are invoking Green’s theorem. Thus:

(ii) If was a conservative field then then the line integral over all closed curves would have to be . But in the previous question we found one closed curve whose line integral is not . Thus is not a conservative field.

The exercise can also be found in the Jom Forum here.

## Existence of constant (2)

Let be a continous function such that and

(1)

Prove that there exists a such that

**Solution**

Now let us consider the function . It holds that . As we can also see using two consecutive DeL’ Hospital’s Rules , it also holds that . So, by Rolle’s theorem there exists a such that

However integration by parts reveals that

and thus which is the desired output.

## Existence of constant (1)

Let be a continous function such that

(1)

Prove that there exists a such that

**Solution**

because is of the form . Thus

due to the initial assumptions. Applying the Integral Mean Value Theorem we have that there exists an such that

Thus . The conclusion now follows from Rolle’s theorem.

## An inequality with integrals

Let be a continuous function. Prove that

**Solution**

The function is concave. Jensen’s inequality in its integral form states that

whenever is concave. Taking the result follows immediately.

## Constant function

Let be a continous function such that and

(1)

Prove that forall .

**Solution**

Clearly as we can see it holds that forall . Also:

(2)

Thus gives us:

Therefore . Thus forall .

The exercise can also be found in mathematica.gr