MATH 251 – LECTURE 25
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
Definition 1. A vector field F is conservative if there exists a function f such that F = ∇f .
Exercise 2. Is F (x, y) = h2x − 6y, −6x + 14y − 5i conservative?
Conservative vector fields
Theorem 3. A vector field F = hP, Qi is conservative if and only if
∂Q
∂P
=
.
∂y
∂x
Conservative vector fields
Exercise 4. Is the vector field F (x, y) = hyex + sin(y), ex + x cos(y)i conservative?
Conservative vector fields
Theorem 5. Let F = ∇f be a conservative vector field. Let C be a path such that ∇f (and f ) is
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 6. 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 7. Let F = ∇f be a conservative
vector field. Let D be a simply connected domain such that
R
F (and 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
Conservative vector fields
Theorem 8. Let F = ∇f be a conservative vector field. Let D be a simply connected domain such that
F (and f ) is continuous in D. Let C be a closed path contained in D. Then,
Z
∇f · dr = 0.
C
Conservative vector fields
Exercise 9. 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
Exercise 10. Let F =
D
y
x
− x2+y
2 , x2 +y 2
E
. Show that F is conservative. Sketch F .
Conservative vector fields
Exercise 11. Let F =
once counterclockwise.
D
y
x
− x2+y
2 , x2 +y 2
E
R
. Evaluate the line integral C F · dr where C is the unit circle traversed
Conservative vector fields
Exercise 12. Let F =
D
y
x
− x2+y
2 , x2 +y 2
E
. Find a function f such that F = ∇f .