c Dr Oksana Shatalov, Spring 2014
1
Spring 2014 Math 251
Week in Review 9
courtesy: Oksana Shatalov
(covering Sections 14.1-14.3 )
14.1-14.2: Vector Fields. Line Integrals
Key Points
• Function u = f (x, y, z) is also called a scalar field. Its gradient is also called gradient vector field.
• Let C be a space curve with parametric equations: x = x(t), y = y(t), z = z(t),
x(t)i + y(t)j + z(t)k, a ≤ t ≤ b. The line integral of f along C is
Z
Z
f (x, y, z) ds =
C
Here ds = |r0 (t)| dt =
a ≤ t ≤ b, or r(t) =
b
f (x(t), y(t), z(t))|r0 (t)| dt.
a
q
2
2
2
[x0 (t)] + [y 0 (t)] + [z 0 (t)] dt.
• Let F be a continuous vector field defined on a curve C given by a vector function r(t), a ≤ t ≤ b. Then the
line integral of F along C is
Z
Z b
F · dr(t) =
F(r(t)) · r0 (t) dt.
C
a
Note that this integral depends on the curve orientation.
Z
•
F · dr(t) = the work done by the force F in moving a particle along a curve C.
C
1. Find the gradient vector field of f (x, y, z) = x ln(y 4 + z 2 )
2. Let C be the line segment starting at (0, 1, 1) and ending at (3, 1, 4). Find the mass of a
thin wire in the shape of C with the density ρ(x, y) = x + y.
2
3. Find the mass of
in the shape of C with the density ρ(x, y, z) = 7y z if C is
a thin wire
2 3
given by r(t) =
t , t, t2 , 0 ≤ t ≤ 1.
3
4. Find the line integral of the vector field F(x, y, z) = h−yz 2 , xz 2 , z 3 i around the circle r(t) =
h2 cos t, 2 sin t, 8i.
5. Find the work done by the force field F(x, y) = 5 + y, − 31 x on a particle that moves along
the curve y = x3 from (−1, −1) to the point (1, 1).
6. A particle moves along the curve C : ~r(t) = ht3 , t2 , ti from the point (1, 1, 1) to the point
(8, 4, 2) due to the force F~ (x, y, z) = hz, y, xi. Find the work done by the force.
c Dr Oksana Shatalov, Spring 2014
2
14.3: The fundamental Theorem for Line Integrals
Key Points
• A vector field F is called a conservative vector field if it is the gradient of some scalar function f s.t
F = ∇f. In this situation f is called a potential function for F.
• The fundamental Theorem for Line Integrals: Let C be a smooth curve given by r(t), a ≤ t ≤ b. Let
f be a differentiable function of two or three variables and ∇f is continuous on C. Then
Z
∇f · dr = f (r(b)) − f (r(a)).
C
• Let F(x, y) = P (x, y)i + Q(x, y)j be a vector field on an open simply connected domain D. Suppose that P
and Q have continuous partial derivatives through D. Then the facts below are equivalent.
–
The field F is
conservativeon D
–
The field F is
conservative on D
⇐⇒
–
The field F is
conservative on D
⇐⇒
∂Q
∂P
=
throughout D
∂x
∂y
–
The field F is
conservative on D
⇐⇒
I
⇐⇒
There exists f s.t. ∇f = F
Z
^
F · dr is independent of path in D
AB
F · dr = 0 for every closed curve C in D
C
7. Find a scalar function f (x, y, z) such that ∇f =< 2xy + z, x2 − 2y, x > and f (1, 2, 0) = 3.
Z
3 4
4 3
~
F~ · d~r where
8. Let F (x, y) = hx y , x y i. Compute
C
a) C is any simple closed path.
b) C is any path from the point M (0, 0) to the point N (1, 2).
9. Let F~ (x, y) = hx + y 2 , 2xy + y 2 i.
a) Show that F~ is conservative vector field.
Z
b) Compute
F~ · d~r where C is any path from (-1,0) to (2,2).
C
10. Let F(x, y) = h2x + y 2 + 3x2 y, 2xy + x3 + 3y 2 i.
(a) Show that F is conservative vector field.
Z
(b) Evaluate
F · dr where C is the arc of the curve y = x sin x from (0, 0) to (π, 0).
C
(Hint: use the previous part.)