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.
Find a potential function of the vector field.
F(x,y) = (3x²y + 1, x³+2y)
Do not guess, show all steps.
Determine whether the line integral of each vector field (in blue) along the semicircular, oriented path (in red) is positive, negative, or zero.
Positive
Positive
Zero
Zero
Negative
Positive
-
1.
Chapter 16 Solutions
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