c Dr Oksana Shatalov, Spring 2013
1
Spring 2013 Math 251
Week in Review 9
courtesy: Oksana Shatalov
(covering Sections 14.1-14.4 )
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 2013
2
14.3: The fundamental Theorem for Line Integrals
14.4: Green’s Theorem
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
• GREEN’s THEOREM: Let C be a positively oriented, piecewise-smooth, simple closed curve in the plane
and let D be the region bounded by C. If P (x, y) and Q(x, y) have continuous partial derivatives on an open
region that contains D, then
I
ZZ ∂Q ∂P
P dx + Q dy =
−
dA.
∂x
∂y
∂D
D
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
~
8. Let F (x, y) = hx y , x y i. Compute
F~ · d~r where
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.)
I
11. Compute the integral I =
(cos x4 + xy)dx + (y 2 ey + x2 )dy , where C is the triangular
C
curve consisting of the line segments from (0, 0) to (1, 0), from (1, 0) to (1, 3), and from
(1, 3) to (0, 0).
12. Compute the integral along the given positively oriented curve C:
Z
(y 2 − arctan x) dx + (3x + sin y) dy,
C
where C is the boundary of the region enclosed by the parabola y = x2 and the line y = 4.
c Dr Oksana Shatalov, Spring 2013
13. Compute the integral
Z
3
(12 − x2 y − y 3 + tan x) dx + (xy 2 + x3 − ey ) dy
C
where C is positively oriented boundary of the region enclosed by the circle x2 + y 2 = 4.
Sketch the curve C indicating the positive direction.
R
14. Given the line integral I = C 4x2 y dx − (2 + x) dy where C consists of the line segment
from (0, 0) to (2, −2), the line segment from (2, −2) to (2, 4), and the part of the parabola
y = x2 from (2, 4) to (0, 0). Use Green’s theorem to evaluate the given integral and sketch
the curve C indicating the positive direction.