Problems: Extended Green’s Theorem
y dx − x dy
exact? If so, find a potential function.
y2
1
x
Answer: M = and N = − 2 are continuously differentiable whenever
y
y
y = 0, i.e. in the two half-planes R1 and R2 – both simply connected.
Since My = −1/y 2 = Nx in each half-plane the field is exact where it is
defined.
To find a potential function f for which F = df we use method 2.
fx = 1/y ⇒ f = x/y + g(y).
fy = −x/y 2 + g / (y) = −x/y 2 ⇒ g / (y) = 0 ⇒ g(y) = c.
⇒ f (x, y) = x/y + c.
1. Is F =
(continued)
1
y
R1
x
R2
Example 3: Let F = rn (xi + yj). Use extended Green’s Theorem to show that F is
conservative for all integers n. Find a potential function.
First note, M
= r
n x, N = r
n y
⇒
My = nr
n−2 xy = Nx ⇔
curlF = 0.
We show F is conservative by showing
C F · dr = 0 for all simple closed curves C.
If C1 is a simple closed curve not around 0 then Green’s Theorem implies C1 F · dr = 0.
F · dr =
If C3 is a circle centered on (0,0) then, since F is radial
C3
C3
F · T ds = 0.
y
C1
If C3 completely surrounds C2 then extended Green’s Theorem
F · dr =
implies
Thus
C2
R
C3
F · dr = 0.
C3
x
F · dr = 0 for all closed loops ⇒ F is conservative.
C
C2
To find the potential function we use method 1 over the curve C shown.
The calculation works for n = 2. For n = 2 everything is the same except we’d get natural
logs instead of powers. (We also ignore the fact that if (x1 , y1 ) is on the negative x-axis
we shoud use a different path that doesn’t go through the origin. This isn’t really an issue
since we already know a potential function exists, so continuity would handle these points
without using an integral.)
1
f (x1 , y1 ) =
r
n x dx + rn y dy
1Cy1
1 x1
2 n/2
=
(1 + y ) y dy +
(x2 + y12 )n/2 x dx
1
=
=
=
(1 +
y 2 )(n+2)/2
n+2
y12 )(n+2)/2
y1
1
+
1
(x2
2(n+2)/2
−
+
n+2
(x12 + y12 )(n+2)/2 − 2(n+1)/2
n+2
(1 +
y
x1
+ y12 )(n+2)/2
n + 2
(x21
+
1
y12 )(n+2)/2
− (1 +
(n + 2)/2
C2
y12 )
(x1 , y1 )
C1
(1, 1)
x
rn+2
+ C.
n+2
If n = −2 we get f (x, y) = ln r + C.
⇒ f (x, y) =
2
MIT OpenCourseWare
http://ocw.mit.edu
18.02SC Multivariable Calculus
Fall 2010
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