I'm specifically referring to this example from my lecture notes:
I understand, algebraically, what's going on, but what does the fact that, for example, $\frac{dx}{dt}>0$ and $\frac{dy}{dt}<0$ for $x>0$ actually mean, geometrically?
The way I interpet it is as follows: e.g. for $x>0$, $\frac{dy}{dx}=\frac{dy}{dt}/\frac{dx}{dt}>0$, but this contradicts the picture, which suggests that, for $x>0$, $\frac{dy}{dx}<0$.
Could someone give me some intuition about this, and how to determine the orientation (i.e. clockwise or anticlockwise) of the spiral using the method in the above notes, because I really don't understand it as it's presented in the notes? Thanks
$\frac{dx}{dt}$ is the derivative of $x$ with regard to time. If it is negative, it means that as time goes on, the value of $x$ is decreasing at the current point, meaning the curve $(x(t),y(t))$ is moving towards the left (where smaller values of $x$ lie), if it is positive, then the curve is moving to the right.
Same with the derivative of $y$ which tells you the up-down movement.
EDIT:
There is no 'time axis' here. What you have is a curve, parametrised by $(x(t), y(t))$ which is a curve in a two-dimensional space. Each value of $t$ gives you one specific value of $x(t)$ and $y(t)$, and together, they give you a point in $\mathbb R^2$. As you look at all of the values of $(x(t), y(t))$ for all time points, the result is a line.
Perhaps the best way of thinking about this is that you have a small ball, for which you know where it is at each time. Now imagine that time passes on and the ball starts moving along its path, its $x$ coordinate changing according to $x(t)$ and its $y$ coordinate according to $y(t)$. If the ball was the tip of a ballpoint pen, it would drag a line behind it. This line is then said to be parametrised by $x(t)$ and $y(t)$, and for a given time $t_0$, the value $\frac{dx}{dt}(t_0)$ tells you how quickly the ball was moving left or right at the time $t_0$.