c Dr Oksana Shatalov, Spring 2013
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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 )
c Dr Oksana Shatalov, Spring 2013
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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
a thin wire
in the shape of C with the density ρ(x, y, z) = 7y z if C is
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.
c Dr Oksana Shatalov, Spring 2013
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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
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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.
8. Let F~ (x, y) = hx y , x y i. Compute
3 4
4 3
Z
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).
c Dr Oksana Shatalov, Spring 2013
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9. Let F~ (x, y) = hx + y 2 , 2xy + y 2 i.
a) Show that F~ is conservative vector field.
Z
F~ · d~r where C is any path from (-1,0) to (2,2).
b) Compute
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
F · dr where C is the arc of the curve y = x sin x from (0, 0) to (π, 0).
(b) Evaluate
C
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).
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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.
13. Compute the integral
Z
(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.
c Dr Oksana Shatalov, Spring 2013
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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.