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14. VECTOR CALCULUS
Problem (Page 996 #36). Determine whether
F(x, y) = hy cos x, sin x
yi
is conservative, and if so, find its potential function.
(We proceed in a manner di↵erent from the text.) This is the same as determining whether
y| cos
{z x} dx + sin
| x
{z y} dy = 0
?
?
M =fx
N =fy
is exact and finding its solution as we did in Math 231.
My = cos x and Nx = cos x =) My = Nx =)
the DE is exact =) F is conservative.
Z
Z
f (x, y) = y cos x dx
f (x, y) = (sin x y) dy
y2
= y sin x
+ h(x)
2
fx(x, y) = y cos x + h0(x)
= y cos x =)
= y sin x + g(y)
fy (x, y) = sin x + g 0(y)
= sin x y =)
0
g (y) =
Thus
y =) g(y) =
y2
+C
2
h0(x) = 0 =) h(x) = C
y2
f (x, y) = y sin x
+C
2
Maple. See vfield(14.1).mw or vfield(14.1).pdf.