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.
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).
Use Cauchy-Riemann equations to
find all points z such that f is
differentiable:
(a) f(z)=
(b) f(z) =|z|+ iz
University Calculus: Early Transcendentals (4th Edition)
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