The integral curves of the timelike unit geodesic vector field \ vec { X } = \ vec { e } _ 0 give the world lines of our observers.
12.
Killing vector field whose integral curves are closed spacelike curves ( circles, in fact ), which moreover degenerate to zero length closed curves on the axis R = 0.
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In physics, integral curves for an electric field or magnetic field are known as field lines, and integral curves for the velocity field of a fluid are known as orbits.
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In physics, integral curves for an electric field or magnetic field are known as field lines, and integral curves for the velocity field of a fluid are known as orbits.
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It is solved by defining certain " kinematical quantities " which completely describe how the integral curves in a congruence may converge ( diverge ) or twist about one another.
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Assume furthermore that \ mu ^ {-1 } ( \ epsilon ) is fibered in circles, each of which is an integral curve of the induced Hamiltonian vector field.
17.
If the vector field X is nowhere zero then it defines a one-dimensional subbundle of the tangent bundle of M, and the integral curves form a regular foliation of M.
18.
One begins by noting that an arbitrary smooth vector field X on a manifold M defines a family of curves, its integral curves u : I \ to M ( for intervals I ).
19.
Thus, the problem of proving the existence and uniqueness of integral curves is the same as that of finding solutions to ordinary differential equations / initial value problems and showing that they are unique.
20.
The integral curves or fibers respectively are certain " pairwise linked " great circles, the orbits in the space of unit norm quaternions under left multiplication by a given unit quaternion of unit norm.
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