Partial Derivatives Instead of Implicit

[Jameson]

On MathWorld's site, they said that

$$(\frac{\partial{y}}{\partial{x}}){_f} = -\frac{(\frac{\partial{f}}{\partial{x}})_{y}}{(\frac{\partial{f}}{\partial{y}})_{x}}$$

So can this method be used instead of implicit differentiation? Will I get the same result? This seems kind of like a parametrics process if this is true.

Thanks,
Jameson

 

[dextercioby]

Yep,that's the formula from the statement of the theorem of implicit functions.

Daniel.

 

[whozum]

Is that for f(x,y)? If so isn't the right hand side = -1?

y(f,x)?

 

[quetzalcoatl9]

yes, and there is nothing mysterious about this equation, btw:

for $f(x,y) = 0$

$$df = \frac{\partial{f}}{\partial{x}}dx + \frac{\partial{f}}{\partial{y}}dy = 0$$

$$\frac{\partial{f}}{\partial{x}}dx = -\frac{\partial{f}}{\partial{y}}dy$$

$$\frac{dy}{dx} = -\frac{\frac{\partial{f}}{\partial{x}}}{\frac{\partial{f}}{\partial{y}}}$$

 

[whozum]

Quetzalcoatl9 I'm not sure if you addressed my question. Isnt there a theorem that says the double partial of a function with respect to alternating variables are equal?

$$f(x,y), f_{xy} = f_{yx}$$ Right? Isn't that what's going on here?

 

[quetzalcoatl9]

Quetzalcoatl9 I'm not sure if you addressed my question. Isnt there a theorem that says the double partial of a function with respect to alternating variables are equal?

$$f(x,y), f_{xy} = f_{yx}$$ Right? Isn't that what's going on here?

I don't understand what you are asking, whozum...do you mean to ask if partial derivatives commute?

 

[whozum]

I don't know what you mean by commute, but please verify that

$$For \ f(x,y), \ f_{xy} = f_{yx}$$

If so, then please explain why the right hand side of the original OP's equation isn't just -1

 

[Gza]

If so, then please explain why the right hand side of the original OP's equation isn't just -1

where'd you get the idea the right side was -1?

 

[whozum]

$$\left(\frac{\delta f}{\delta x}\right)_y = f_{xy}$$

$$\left(\frac{\delta f}{\delta y}\right)_x = f_{yx}$$

$$f_{yx} = f_{xy}$$

edit: I'm obviously missing something, I just want someone to point out what it is.

 

[dextercioby]

Yes,you are.Missing something,that is.We use this notation

$$\left (\frac{\partial f}{\partial x}\right)_{y}\equiv \frac{\partial f(x,y)}{\partial x}$$

,that is we explain which variables are kept constant during the partial differentiaition,viz. "y" in this case.

This notation,or convention,if u prefer,is very common in thermodynamics.

Daniel.

 

[whozum]

So that hanging y on the LHS is just to indicate that its constant, then. I had that confused as a derivative.

 

[arildno]

The notation is excellent in a field like thermo-dynamics, where it is convenient to switch between which quantities are to be regarded as independent quantities and which are to be regarded as dependent quantities.

It is rather redundant in a field where there exisst a "natural" choice of the independent variables.

 

[dextercioby]

To give u an idea

$$\frac{\partial F}{\partial V}$$

,where F is the free energy/Helmholtz potential & V is the volume,doesn't mean anything in thermodynamics.

Daniel.

 

[arildno]

Are you talking to me??

 

[dextercioby]

Of course,not.I gave that example to Whozum...

Hmmmmm...

Daniel.

 

[quetzalcoatl9]

Are you talking to me??

...<in bronx-accented Al Pacino voice>

 

[arildno]

I haven't bothered to get a driving license yet.
Perhaps I should..