Let $u = u(x,y)$ be a scalar fields on $\mathbb R^2$. Consider the linear first order PDE for $u$,$$(x\partial_y - y\partial_x)u = 0$$ (i) Rewrite $\partial_{\theta}u = 0$ as a directional derivative
(ii) Determine the curves along which $u$ = constant
For (i) so far I have tried the following:
Since the directional derivative is the rate of change of a function in this case
which is $u = (x,y)$ in the direction of a unit vector $\vec d = <a,b>$.
Lets define a new function of a single variable $$g(z) = u(x_0 + az, y_0 + bz)$$
where $x_0$, $y_0$, a and b are some fixed numbers.
Then by the definition of the derivative of a functions of a single variable we have $$g'(z) = \lim_{h->0} \frac{g(z+h)-g(z)}{h}$$
and the derivative of a function at $z=0$ is given by
$$g'(0) = \lim_{h->0} \frac{g(h)-g(0)}{h}$$
$g'(0) = D_{\vec d}u(x_0,y_0)$.
From here I have tried to define a new function respect to two variables x and y
but really stuck. It seems like I have to solve(i) to proceed to (ii). Any
help to solve this will be appreciated.
You are supposed to write $x(\theta)$, $y(\theta)$ where $\theta$ parameterizes the curve along which you take the derivative. Then calculate $$\frac{d u(x(\theta),y(\theta))}{d\theta}$$ and determine what equation $x(\theta)$ and $y(\theta)$ have to fulfill in order that $du/d\theta$ corresponds to $(x\partial_y -y\partial_x)u$.
Edit: Due to the comment of the OP.
The directional derivative along the curve $\gamma=(x(\theta), y(\theta))$ is simply defined as $$ \frac{d u(\gamma)}{d\theta} = x'(\theta) \partial_xu + y'(\theta) \partial_y u. $$ This is essentially what you wrote in your questions. Now we want to have that $$ x' = -y, \quad y' =x \tag{1}$$ such that $$\frac{du}{d\theta}=(x\partial_y -y\partial_x)u =0,$$ i.e., $u$ is constant along the curve $\gamma$. So you have to find the general solution to (1) to find the curves along which $u$ is constant. Along each curve you may choose a different constant such that at the end you can set a function on a set of points "perpendicular" to the family of curves as initial condition.