Find the value of the line integral.F · dr (Hint: If F is conservative, the integration may be easier on an alternative path.) F(x,y) = ye^xyi + xe^xyj (a) r1(t) = ti − (t − 8)j, 0 ≤ t ≤ 8 ?
(b) the closed path consisting of line segments from (0, 8) to (0, 0), from (0, 0) to (8, 0), and then from (8, 0) to (0, 8)
(b) the closed path consisting of line segments from (0, 8) to (0, 0), from (0, 0) to (8, 0), and then from (8, 0) to (0, 8)
1 Answer
Zero in both cases
Explanation:
A field
bbF = - nabla phiF=−∇ϕ
Or, in 2-D:
bbF = - (: phi_x, phi_y:) F=−⟨ϕx,ϕy⟩
This can be solved directly here, because:
So for:
Path a)
bbr_1(t) = t bbi − (t − 8) bbj, qquad 0 le t le 8
This is along line
Path b)
Regardless of path, this is closed path from
Going instead with the hint , the curl of a conserved vector field is zero. This follows from the existence of a potential function and the gradient theorem.
Curl of the Field:
The field is conservative / path-independent
Path a)
This is along line
Following the hint, break this into
{(bb1, x = 0, dx = 0, y in [8,0]),(bb2, y = 0, dy = 0, x in [0,8]):}
Path
Path
Path b)
This is clearly zero as it is a closed path in a conserved field.
FINALLY:
Noting that
bbr(t) : {(x = t),(y = 8 - t):} qquad bbr'(t) : {( x' = 1),( y' = -1):}
