Evaluating a Line Integral of a Vector Field Using Technology In Exercises 35 and 36, use a computer algebra system to evaluate ∫ c F · d r . F ( x , y , z ) = x 2 z i + 6 y j + y z 2 k C : r ( t ) = t i + t 2 j + ln t k , 1 ≤ t ≤ 3
Evaluating a Line Integral of a Vector Field Using Technology In Exercises 35 and 36, use a computer algebra system to evaluate ∫ c F · d r . F ( x , y , z ) = x 2 z i + 6 y j + y z 2 k C : r ( t ) = t i + t 2 j + ln t k , 1 ≤ t ≤ 3
Solution Summary: The author explains how to calculate the line integral displaystyle 'underset' Cint's F.dr over the curve.
Evaluating a Line Integral of a Vector Field Using Technology In Exercises 35 and 36, use a computer algebra system to evaluate
∫
c
F
·
d
r
.
F(x, y, z) =
x
2
z
i
+
6
y
j
+
y
z
2
k
C
:
r
(
t
)
=
t
i
+
t
2
j
+
ln
t
k
,
1
≤
t
≤
3
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.
Rain on a roof Consider the vertical vector field F = ⟨0, 0, -1⟩, correspondingto a constant downward flow. Find the flux in the downward direction acrossthe surface S, which is the plane z = 4 - 2x - y in the first octant.
Maximum curl Let F = ⟨z, x, -y⟩.a. What is the scalar component of curl F in the direction of n = ⟨1, 0, 0⟩?b. What is the scalar component of curl F in the direction ofn = ⟨0, -1/√2, 1/√2⟩?c. In the direction of what unit vector n is the scalar componentof curl F a maximum?
Differentiation of Vector-Valued Functions In Exercises
7 and 8, find r (t), r(t,), and r (t,) for the given value of t. Then
sketch the space curve represented by the vector-valued
function, and sketch the vectors r(t) and r'(t).
7. r(t) = 2 cos ti + 2 sin tj + tk, to
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