(a) Evaluate the line integral ∫ C F ⋅ d r , where F ( x , y , z ) = x i − z j + y k and C is given by r ( t ) = 2 t i + 3 t j − t 2 k , − 1 ≤ t ≤ 1 . (b) Illustrate part (a) by using a computer to graph C and the vectors from the vector field corresponding to t = ± 1 and ± 1 2 (as in Figure 13).
(a) Evaluate the line integral ∫ C F ⋅ d r , where F ( x , y , z ) = x i − z j + y k and C is given by r ( t ) = 2 t i + 3 t j − t 2 k , − 1 ≤ t ≤ 1 . (b) Illustrate part (a) by using a computer to graph C and the vectors from the vector field corresponding to t = ± 1 and ± 1 2 (as in Figure 13).
Solution Summary: The author evaluates the line integral of F along C by using the power rule of differentiation.
(a) Evaluate the line integral
∫
C
F
⋅
d
r
, where
F
(
x
,
y
,
z
)
=
x
i
−
z
j
+
y
k
and C is given by
r
(
t
)
=
2
t
i
+
3
t
j
−
t
2
k
,
−
1
≤
t
≤
1
.
(b) Illustrate part (a) by using a computer to graph C and the vectors from the vector field corresponding to
t
=
±
1
and
±
1
2
(as in Figure 13).
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
How do you graph the vector field F = ⟨ƒ(x, y), g(x, y)⟩?
A net is dipped in a river. Determine the
flow rate of water across the net if the
velocity vector field for the river is given
by v=(x-y,z+y+7,z2) and the net is
decribed by the equation y=1-x2-z2, y20,
and oriented in the positive y- direction.
(Use symbolic notation and fractions
where needed.)
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