Flow curves in the plane Let F ( x , y ) = ( f ( x , y ) , g ( x , y ) ) be defined on ℝ 2 . 54. Find and graph the flow curves for the vector field F = 〈 y , x 〉. Note that d / d x ( y 2 ) = 2 y y ′ ( x )
Flow curves in the plane Let F ( x , y ) = ( f ( x , y ) , g ( x , y ) ) be defined on ℝ 2 . 54. Find and graph the flow curves for the vector field F = 〈 y , x 〉. Note that d / d x ( y 2 ) = 2 y y ′ ( x )
Solution Summary: The author analyzes the flow curve of the vector field F=langle y,xrangle and graphs it.
Flow curves in the planeLet
F
(
x
,
y
)
=
(
f
(
x
,
y
)
,
g
(
x
,
y
)
)
be defined on
ℝ
2
.
54. Find and graph the flow curves for the vector field F = 〈y, x〉. Note that
d
/
d
x
(
y
2
)
=
2
y
y
′
(
x
)
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.
The figure shows the vector field
F (x,y) = {2xy,x2 } and three curves that start at (1,2) and end at (3,2).
(a) Explain why∫c F. dr has the same value for all the three curves.
(b) What is this common value?
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