Flow curves in the plane Let F ( x , y ) = ( f ( x , y ) , g ( x , y ) ) be defined on ℝ 2 . 51. Explain why the flow curves or streamlines of F satisfy y ′ = g ( x , y ) / f ( x , y ) and are everywhere tangent to the vector field.
Flow curves in the plane Let F ( x , y ) = ( f ( x , y ) , g ( x , y ) ) be defined on ℝ 2 . 51. Explain why the flow curves or streamlines of F satisfy y ′ = g ( x , y ) / f ( x , y ) and are everywhere tangent to the vector field.
Flow curves in the planeLet
F
(
x
,
y
)
=
(
f
(
x
,
y
)
,
g
(
x
,
y
)
)
be defined on
ℝ
2
.
51. Explain why the flow curves or streamlines of F satisfy
y
′
=
g
(
x
,
y
)
/
f
(
x
,
y
)
and are everywhere tangent to the vector field.
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 temperature on a cubic box [0, 4] × [0, 4] × [0, 4] (measured in meters) can be describedby the function T (x, y, z) = x2y + y2z degrees F◦. A fly is in position (1, 2, 1) and takesoff in a straight line to the corner (4, 0, 4). Use directional derivatives to calculate the changein temperature the fly experiences as she takes off. Give your answer with 2 decimal digitscorrect.
2. Calculate the gradient vector Vf of the function f (x, y) = x² – x + y - x²y - 2y2 at
the point (2,1) and sketch it on the attached contour plot (you can save the picture, open
in photo editor and use drawing tools).
Explain in one paragraph (about 200-300 words) the meaning of the gradient vector
Vf(2,1), negative gradient vector -Vf(2,1).
(a) Evaluatef(2,1).(b) Estimatefx(2,1) andfy(2,1) using the contour plot.(c) Determine the linearizationL(x, y) at the point (2,1)
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University Calculus: Early Transcendentals (4th Edition)
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