## 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

Let and . Integrating by parts reveals that

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

Let be a primitive of . Consider the function . Trivially . Now, we note that:

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

Consider the function   which is differentiable in . We can easily see that has a global minimum at that is equal to . Visually we have that:

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