(2) Calculus III (Vector Fields)... Find the value of the given line integral. Give exact answers only...

THIS IS A 2 PART QUESTION.

Find the value of the given line integral. Give exact answers only. Hint: If the vector field is conservative, then it will be easier to evaluate using the Fundamental Theorem of Line Integrals. If the field is not conservative then one of two possibilities:
1) If there is not enough information to do problem, then state so. That is, write: CANT DO IT as answer.
2) If there is enough info to complete the line integral, go ahead and evaluate it by BRUTE FORCE, that is without greens theorem, u must parameterize first.

OK here are the questions:

(a) ∫c F·dr for F(x,y,z) = < (e^x cos(y)), (-e^x sin(y)), (e^x cos(z)) >
where the curve C starts from the point (1,0,1) and goes to the point (1, pi/2 , 0).


(b) ∫c F·dr for F(x,y,z) = < Ln(x) + sec²(x+y) , (y / (y² + z²))+sec²(x+y) , (z / (y² + z²)) > and where the
curve C starts from the point (1,0,1) and goes to the point (1,-1,1)

(a) Note that curl F = <0, -e^x cos z, 0>.
Since this is not zero, the integral is not independent of path.
So, we can't do this integral as presented.

(b) This time, curl F = <0, 0, 0>.

Integrating each entry of F to find a potential function f,
f = ∫ (ln x + sec^2(x + y)) dx = x ln x - x + tan(x + y) + r(y, z)
f = ∫ (y/(y^2 + z^2) + sec^2(x + y)) dy = (1/2) ln(y^2 + z^2) + tan(x + y) + s(x, z)
f = ∫ (z/(y^2 + z^2)) dz = (1/2) ln(y^2 + z^2) + t(x, z).

Hence, f(x, y, z) = x ln x - x + tan(x + y) + (1/2) ln(y^2 + z^2).

So, FTC for line integrals yields
[x ln x - x + tan(x + y) + (1/2) ln(y^2 + z^2)] {for (x,y,z) = (1,0,1) to (1,-1,1)}
= tan 1.

I hope this helps!