Flow of a vector field?

Question

How can I find a flow curve for a vector field given as,

$$U(x,y,z)=(2z+x, y-z,z+y)$$

with a condition that $r(0)=(x_0,y_0,z_0)$.

Answer 1

Write down $$\frac{d}{dt}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix} 2z+x\\ y-z \\ z+y \end{bmatrix}=\begin{bmatrix} 1 & 0 &2 \\ 0&1&-1 \\ 0&1&1\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}$$ Formally you're done, as $d\vec{r}/dt=A\vec{r}$ is solved by $\vec{r}=e^{At}\vec{r}_0$, but you probably want to express it a bit less symbolically than as a matrix exponential.

You can resolve this by finding the eigenframe of the matrix, which collapses the problem into 3 independent equations.