Integral curve, flow

We define a specific vector field U. The differential equations for the integral curve of U are

Both of the following lines do the same, namely solve an integral curve α:I→ numerically. The first curve starts at the initial point α(0)={1,0}. The curve is parametrized on the interval I={-1,1}.

We plot the vector field and the integral curves.

The unit circle gets transported by the flow of U below.

We compute the differential of the flow ψ(t,x) of a vector field V along the integral curve with initial point ∈M. The differential depends on the parameter t∈I. Below, we chose ={1,1} and I={-1,1}.

Here comes the picture.

Created by Mathematica (December 22, 2006) |