If I have a vector field (x,y,z) on a one dimensional line (x-axis) and I have to check its monotonicity between two intervals.

Carsen Patel 2022-08-13 Answered
Monotonicity of vector fields
If I have a vector field (x,y,z) on a one dimensional line (x-axis) and if I have to check its monotonicity between two intervals. Will it be monotonic if:
(1) only one of the components of vector field is monotonic in that interval. OR (2) two of the components of vector field are monotonicaly increasing and the other is monotonicaly decreasing in that interval. OR (3) all the components of vector field are either monotonicaly increasing or monotonicaly decreasing in that interval. .
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Answers (1)

Cynthia George
Answered 2022-08-14 Author has 10 answers
Explanation:
If I is your interval, and for each t I your vector field at t is (x(t),y(t),z(t)), all you need to do is check that for each t I
x ( t ) < y ( t ) < z ( t ) ,,
or the analogous inequalities with > instead of <. These conditions can be met regardless of whether each individual coordinate function is increasing, decreasing or not monotonic at all.
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