(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.
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