91Ó°ÊÓ

Rectify the direction fields of the equations \(\dot{x}=t\) and \(\dot{z}=x^{2}\) in a neighborhood of the origin.

Is every smooth direction field in the plane globally rectifiable?

.\( Is it possible to rectify the direction field of the equation \)\dot{x}=v(t, x)\( on the whole extended phase space \)\boldsymbol{R} \times \boldsymbol{R}^{n}$ when the right-hand side is smooth and defined on this entire space?

Solve the Cauchy problem \(\left.u\right|_{x=0}=\sin y\) for the equation \(\partial u / \partial x=\) \(y \partial u / \partial y+y\)

Knowing the components of the fields \(a\) and \(b\) in bome coordinate system, find the components of their commutator.

Let \(\left\\{g^{t}\right\\}\) be the phase flow of the field \(a\) and \(\left\\{h^{*}\right\\}\) the phase flow of the field \(\boldsymbol{b}\). Prove that the flows commute \(\left(g^{t} h^{\prime} \equiv h^{\circ} g^{t}\right)\) if and only if the commutator of the fields is zero.

Find the derivative of the solution of the pendulum equation \(\bar{\theta}=-\sin \theta\). with initial condition \(\theta(0)=a, \dot{\theta}(0)=0\) with respect to \(a\) at \(a=0\).

Prove that the sectorial velocity is a first integral of the equation \(\overline{\boldsymbol{r}}=\operatorname{ra}(r)\) for any form of the function a. A force field of the form \(\operatorname{raf}(r)\) is called central. The preceding problem shows why the law of universal gravitation cannot be deduced from Kepler's Second Law: the third faw is needed.