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Mathematics
Differential Geometry
Coordinate representation of a diffeomorphism
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[QUOTE="martinbn, post: 5985957, member: 252793"] If I understand correctly your question and notations, I think this is not true. It will not be just integration, you need to solve a system of differential equations. Take any example. Say ##\mathbb R^2## with the usual coordinates ##(x,y)##, let the vector field be ##V=x^2\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}##. Then the parametrized path you are looking for satisfies ##\frac{dx}{d\lambda}=x^2## ##\frac{dy}{d\lambda}=y## Solving the equations is not just a matter of integration. Well in a way it is because I chose them simple to solve, but in general the right sides will depend on all coordinate function, they will not be decoupled as in this example. The dependence on the initial point will come from the initial conditions for the equations ##x(0)=x_0## and ##y(0)=y_0##. [/QUOTE]
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Differential Geometry
Coordinate representation of a diffeomorphism
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