MATH 251 – LECTURE 26
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 a conservative vector field. Let C be a path such that ∇f and f are
continuous on C. Let C be parametrized by the vector function r(t) for a ≤ t ≤ b. Then
Z
∇f · dr = f (r(b)) − f (r(a)).
C
Conservative vector fields
Exercise 2. Let F = hyz, xz, xy +14zi. Evaluate the line integral
to (4, 4, 3).
R
C
F ·dr where C is any curve from (1, 0, −1)
Conservative vector fields
Theorem 3. 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.
Conservative vector fields
Theorem 4. 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
Conservative vector fields
Exercise 5. Let F = hy, xi, and let C be the unit circle traversed once counterclockwise. Evaluate the line
integral
Z
F · dr.
C
Conservative vector fields
D
y
Exercise 6. Let F = − x2+y
2,
x
2
x +y 2
E
. Show that F is conservative. Sketch F .
Conservative vector fields
D
Exercise 7. Let F =
once counterclockwise.
y
x
− x2+y
2 , x2 +y 2
E
. Evaluate the line integral
R
C
F · dr where C is the unit circle traversed
Conservative vector fields
D
y
Exercise 8. Let F = − x2+y
2,
x
2
x +y 2
E
. Find a function f such that F = ∇f .
Green’s theorem
Definition 9. 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 10 (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 11. 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 11. 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 12. 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 12. 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