MATH 251 – LECTURE 27
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 14.3–4
webAssign: 14.3–4, due 4/11 11:55 p.m.
Next week: 14.5–6
webAssign: 14.5–6, opens 4/11 12 a.m.
Help Sessions:
Sun–Thu 6–8 p.m. in BLOC 149
Office Hours:
BLOC 641C
M 12:30–2:30 p.m.
W 2–3 p.m.
or by appointment.
Conservative vector fields
Theorem 1. Let F = ∇f be Ra conservative vector field. Let D be a simply connected domain such that
F is continuous on D. Then C F · dr is independent of path in D. That is, if C is the curve from P1 to
P2, then
Z
∇f · dr = f (P2) − f (P1).
C
Theorem 2. Let F = ∇f be a conservative vector field. Let D be a simply connected domain such that
F is continuous in D. Let C be a closed path contained in D. Then,
Z
∇f · dr = 0.
C
Green’s theorem
Definition 3. A closed oriented curve C ⊂ R2 is said to be simple if it has no self-intersections. A simple
oriented curve encloses a simply connected domain D. The curve C is said to be positively oriented if an ant
walking along the curve, in the direction of the orientation, has D on its left side.
Green’s theorem
Theorem 4 (Green’s theorem). Let F = hP, Qi be a vector field, and let C be a simple, closed, positively
oriented curve, enclosing a domain D. Then
Z
ZZ ∂Q ∂P
F · dr =
−
dA
∂x
∂y
C
D
Green’s theorem
Exercise 5. Let C be the curve consisting of the line segments from (0, 0) to (2, 0), from (2, 0) to (0, 2), and
from (0, 2) to (0, 0). Use Green’s theorem to evaluate the line integral
Z
x2y 2 dx + 5x2y dy.
C
Green’s theorem
Exercise 6. Let C be the boundary of the region enclosed by the curves y = x2 and y = 2, with positive
orientation. Evaluate the line integral
Z √ x
arctan y 2
3y + 7e
dx + 8x + 2 cos e
dy.
C
Green’s theorem
Exercise 6. Let C be the boundary of the region enclosed by the curves y = x2 and y = 2, with positive
orientation. Evaluate the line integral
Z √ x
arctan y 2
3y + 7e
dx + 8x + 2 cos e
dy.
C
Green’s theorem
Exercise 7. Compute the area of the disc D = {x2 + y 2 ≤ R2} using Green’s theorem.
Green’s theorem
Exercise 8. Find the area of the region bounded by the curve with vector equation r(t) = hcos(t), sin(2t)i,
where − π2 ≤ t ≤ π2 .