Total derivative -> partial derivative

[Walkingman]

Under what conditions can you replace a total differential with a partial?

dx/dy -> partial(dx/dy)

in the context of 2 independent variables and multiple independent variables.

Thanks

 

[dvs77]

let x(u,v),and u,v are functions of y
then
dx/dy=∂x/∂u.∂u/∂y+∂x/∂v.∂v/∂y

 

[Walkingman]

but under what conditions can i say

dx/dy = ∂x/∂y ?

 

[HallsofIvy]

Just as dvs77 said: precisely when x depends only on y!

However, that was that really your question? You originally asked "Under what conditions can you replace a total differential with a partial?" A "total differential" is not " a derivative". In other words, not dx/dy. In terms of two independent variables, x and y, the total differential of a function f(x,y) is $$df= \frac{\partial f}{\partial x}dx+ \frac{\partial f}{\partial y}dy$$.

Notice that, in the case you are describing, x and y are not reall "independent".

 

[Walkingman]

Thanks

And I was looking for the total derivative, sorry about the mistake in the op.

 

[dextercioby]

There's total derivative and there's total differential...Which one are u after...?Halls gave the simplest example of a (total) differential.

The really interesting case is when dependece upon a variable is both explicite & implicite

$$z=z\left(\frac{x}{t},yt^{2},u(t),t\right)$$

and u want to compute

$$\frac{dz}{dt}$$


Daniel.

 

[kleinwolf]

let x(u,v),and u,v are functions of y
then
dx/dy=∂x/∂u.∂u/∂y+∂x/∂v.∂v/∂y

To be more precise with notations this should be

$$\frac{\partial x}{\partial u}\frac{du}{dy}+\frac{\partial x}{\partial v}\frac{dv}{dy}$$

Note that if z=f(x,y) and y=g(x) then d/dx and \partial_x both exist but they are different.

Another interesting case is : z=f(x,z)...(implicit functions)