Evaluate the function

Question

(a) Find a function f such that and (b) use part (a) to evaluate   along the given curve C.

Answer 1

Step 1:

The function is .

(a)

Find the potential function .

To find the potential function, follow the condition .

From above , and .

Integrate with respect to x.

           equation (1)

Differentiate equation (1) with respect to y.

          equation (2)

Compare equation (2) and .

                                   equation (3)

Integrate equation (3) with respect to y.

Substitute in equation (1)

               equation (4)

Differentiate equation (4) with respect to z.

              equation (5)

Compare equation (5) and .

                                    equation (6)

Integrate equation (6) with respect to z.

Substitute in equation (4).

.

Solution :

(a) .

Answer 2

(b)

Step 1:

Find .

But , then.

In a smooth curve C, vector function in the interval , whose gradient vector

is continuous on C  then.

C is the line segment from .

Then

Solution :

(b) .