Use Stoke's Theorem to evaluate ∫c F *dr. In each case C is oriented counterclockwise

Question

 when viewed from above. F(x,y,z) = e^-xi +e^xj +e^zk? 

 

 

C is the boundary of the part of the plane 2x +y +2z = 2 in the first octant

Answer 1

Given function is

The given plane S = 2x + y + 2z = 2Surface and taking the ground plane as the xy-plane.

So p = k

The given plane S is in the xy - plane and bounded by x axis , y axis.So z coordinate is nullified.

Now given line is 2x + y + 2(0) = 2   ⇒   2x + y  = 2  ⇒   y  = 2 - 2x.

Boundaries are determined as

If y = 0 ⇒   2x + 0  = 2  ⇒    2x  = 2   ⇒   x  = 1.

So  0 ≤ x ≤ 1  and 0 ≤ y < 2 - 2x.

Now substitute the values of n , σ , curl F in stokes theorem.

Solution : ∫_c F · dr = 2e - 4.