Why isn't ∂x/∂y = 1/2(∂x/∂u*∂u/∂y+∂x/∂v*∂v/∂y)?

Question

Why isn't $\frac{\partial x}{\partial y}$ equal to $$\frac{1}{2}\left( \frac{\partial x}{\partial u}\frac{\partial u}{\partial y} + \frac{\partial x}{\partial v} \frac{\partial v}{\partial y} \right)$$ but to $$\frac{\partial x}{\partial u}\frac{\partial u}{\partial y} + \frac{\partial x}{\partial v} \frac{\partial v}{\partial y} $$ and why do we omit the $\frac{1}{2}$ factor? How does the equation still remains balanced?

Answer 1

Try it out in some simple examples.

Answer 2

We consider $x=x(u(y),v(y))$.

Imagine the following situation: the descartes plane with coordinates $u$ and $v$, $y$ will represent the time, and $x$ will be some characteristics of the moving point on that plane.

The time $y$ changed a little - hence the point moved from $(u(y),v(y))$ to $(u(y+\Delta y),v(y+\Delta y))$.

We will represent the movement as two steps: first, we move in the direction of coordinate $u$ (this gives us $\partial x/\partial u$) and take into account the velocity in that direction (this gives us the factor $\partial u/\partial y$ ), then (the sign $+$) we move in the direction of coordinate $v$ (this gives us $\partial x/\partial v$) and take into account the velocity in that direction (this gives us the factor $\partial v/\partial y$ ).

The underling idea - in order to arrive to point $(1,1)$ from $(0,0)$ we first move to $(1,0)$ and only then to $(1,1)$, i.e. we move along axes.