Use a graph of the vector field F and the curve C to guess whether the line integral of F over C is positive, negative or zero. Then evaluate the line integral. F ( x , y ) = x x 2 + y 2 i + y x 2 + y 2 j , C is the parabola y = 1 + x 2 from ( − 1 , 2 ) to ( 1 , 2 )
Use a graph of the vector field F and the curve C to guess whether the line integral of F over C is positive, negative or zero. Then evaluate the line integral. F ( x , y ) = x x 2 + y 2 i + y x 2 + y 2 j , C is the parabola y = 1 + x 2 from ( − 1 , 2 ) to ( 1 , 2 )
Solution Summary: The author explains that the line integral of F over C is positive, negative, or zero by using a graph of the vector field.
Use a graph of the vector field F and the curve C to guess whether the line integral of F over C is positive, negative or zero. Then evaluate the line integral.
F
(
x
,
y
)
=
x
x
2
+
y
2
i
+
y
x
2
+
y
2
j
,
C is the parabola
y
=
1
+
x
2
from
(
−
1
,
2
)
to
(
1
,
2
)
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.
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 + 3, z²) and the net is decribed by the equation y = √1-x²-2², y ≥ 0, and oriented in the positive y-
direction.
(Use symbolic notation and fractions where needed.)
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
= (x - y, z + y + 9, z) and the net is decribed by the equation y = V1-x - z, y 2 0, and oriented in the positive y-
direction.
(Use symbolic notation and fractions where needed.)
V. dS =
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