Total derivative -> partial derivative

In summary, you can replace a total differential with a partial if the dependency on x is both explicite and implicite.
  • #1
Walkingman
8
0
Under what conditions can you replace a total differential with a partial?

dx/dy -> partial(dx/dy)

in the context of 2 independant variables and multiple dependant variables.

Thanks
 
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  • #2
let x(u,v),and u,v are functions of y
then
dx/dy=∂x/∂u.∂u/∂y+∂x/∂v.∂v/∂y
 
  • #3
but under what conditions can i say

dx/dy = ∂x/∂y ?
 
  • #4
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 [tex]df= \frac{\partial f}{\partial x}dx+ \frac{\partial f}{\partial y}dy[/tex].

Notice that, in the case you are describing, x and y are not reall "independent".
 
  • #5
Thanks

And I was looking for the total derivative, sorry about the mistake in the op.
 
  • #6
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

[tex] z=z\left(\frac{x}{t},yt^{2},u(t),t\right) [/tex]

and u want to compute

[tex]\frac{dz}{dt} [/tex]


Daniel.
 
  • #7
dvs77 said:
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

[tex] \frac{\partial x}{\partial u}\frac{du}{dy}+\frac{\partial x}{\partial v}\frac{dv}{dy} [/tex]

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)
 

Related to Total derivative -> partial derivative

1. What is the difference between total derivative and partial derivative?

The total derivative is the derivative of a function with respect to all of its variables, while the partial derivative is the derivative of a function with respect to one variable while holding all other variables constant.

2. How is the total derivative related to the chain rule?

The total derivative can be seen as a generalization of the chain rule, where it takes into account changes in all variables instead of just one.

3. Can the total derivative be written in terms of partial derivatives?

Yes, the total derivative can be expressed as a linear combination of the partial derivatives of the function.

4. Why is the total derivative used in multivariable calculus?

The total derivative provides a way to calculate the rate of change of a function with respect to all of its variables, making it useful in studying systems with multiple variables and their interactions.

5. How does the concept of total derivative extend to higher dimensions?

In higher dimensions, the total derivative becomes the Jacobian matrix, which includes all partial derivatives of a function with respect to all of its variables. This allows for the calculation of higher-order derivatives and more complex systems.

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