A vector field F gives a specific vector F(x, y, z) for each point (x, y, z) in the domain of F. An integral curve or flow line of ♬ is a path t → r(t) for which F(r(t)) = along each t in the domain of the path. In particular, if F = (P, Q, R) and r' = (x, y, z) then dr the integral curve must solve: dz dt where P, Q, R are evaluated at (x(t), y(t), z(t)). Find the integral curves of the vector fields: P = dx dt' Q = dy dt' (a.) Ġ(x, y, z) = (a, b, c) where a, b, c are constants, (b.) F(x, y, z) = (−y, x, 1). R = To solve the differential equations which arise in part (b.) here, you want to eliminate all but one dependent variable (these are x, y, z here) and solve the resulting ordinary differential equation.
A vector field F gives a specific vector F(x, y, z) for each point (x, y, z) in the domain of F. An integral curve or flow line of ♬ is a path t → r(t) for which F(r(t)) = along each t in the domain of the path. In particular, if F = (P, Q, R) and r' = (x, y, z) then dr the integral curve must solve: dz dt where P, Q, R are evaluated at (x(t), y(t), z(t)). Find the integral curves of the vector fields: P = dx dt' Q = dy dt' (a.) Ġ(x, y, z) = (a, b, c) where a, b, c are constants, (b.) F(x, y, z) = (−y, x, 1). R = To solve the differential equations which arise in part (b.) here, you want to eliminate all but one dependent variable (these are x, y, z here) and solve the resulting ordinary differential equation.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:A vector field F gives a specific vector F(x, y, z) for each point (x, y, z) in the domain of
F. An integral curve or flow line of F is a path t → r(t) for which F(r(t)) = dr along
each t in the domain of the path. In particular, if F = (P, Q, R) and r = (x, y, z) then
the integral curve must solve:
dt
P
=
da
dt'
Q
-
dy
dt'
R
(a.) G(x, y, z) = (a, b, c) where a, b, c are constants,
(b.) F(x, y, z) = (−y, x, 1).
=
dt
where P, Q, R are evaluated at (x(t), y(t), z(t)). Find the integral curves of the vector
fields:
To solve the differential equations which arise in part (b.) here, you want to eliminate
all but one dependent variable (these are x, y, z here) and solve the resulting ordinary
differential equation.
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