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QUIZ 7, Version B : MATH 251, Section 506
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Write up your result, detail your calculations if necessary and BOX your final answer
1. Given F (x, y) = hy cos x + sin xi, sin x.
(a) [25pts] Show that F is conservative.
(b) [25pts] Find a potential function of F .
2. [50pts] Use the Green’s Theorem to compute
(0, 1) and (1, 1).
H
C
−xydx+ydy where C is the triangle of vertices (0, 0),
1. F =< P, Q > is a vector field in R2 whose every component function is defined in R2 and has also
continuous first partial derivatives.
(a) Since the domain of F is R2 which is simply connected . We need to check that Qx (x, y) = Py (x, y)
for every (x, y) in R2 :
Py (x, y) = cos x and Qx (x, y) = cos x. So, F is conservative.
(b) Let f be a potential function which satisfies
fx (x, y) = y cos x + sin x,
and fy (x, y) = sin x.
We integrate first fx with respect to x and obtain f (x, y) = y sin x − cos x + g(y) where g is a
differentiable function. So, now we differentiate this function f w.r.t y and we obtain fy (x, y) =
sin x + g 0 (y) which needs to be equal to Q(x, y) = sin x. By identification, we obtain g 0 (y) = 0
which implies g(y) = Cst. Then, a potential function is
f (x, y) = y sin x − cos x ,
(by choosing Cst = 0).
2. Let P (x, y) = −xy and Q(x, y) = y. By Green’s theorem, by denoting D = {(x, y) | 0 ≤ x ≤ 1, x ≤ y ≤ 1}
the domain bounded by C, we have
Z
Z Z
Z Z
−xydx + ydy =
(Qx − Py )dA =
xdA,
C
Z
D
1Z 1
=
xdydx =
0
=
D
Z
1
.
6
x
0
1
x2 x3
x(1 − x)dx =
−
2
3
1
,
0