Total Derivative

[TheLivingMartyr]

I've been looking at some partial differential equation solving recently, and things like the total derivative come up quite often, lets say,

 

When you can constrain y as a function of x, then

 

[math]

\frac{d}{dx} z(x,y) = \frac{dz}{dx} + \frac{dz}{dy} \frac{dy}{dx}

[/math]

 

(didn't know how to get the partial operator symbol, but I'm sure you know where it's meant to be)

 

Now I understand that this obviously works, it can be shown via the chain rule, but I have two little intuitive problems with it

 

1. If you know y as a function of x, why can't you simply substitute in the function of x, and then just have an ordinary differential equation of z(x)

 

2. Surely the "Total Derivative" ought to show how the function varies as all variables vary, so why can't you have a total derivative when all the variables are independent (something like "The derivative of z with respect to x and y").

[mathematic]

For question 1 - your observation is correct, although I wouldn't call it a differential equation. You are simply taking a derivative.

 

For question 2 - if x and y are not connected in any way, then total derivative has no meaning. You just have the partials.

[TheLivingMartyr]

hmm, I see what you mean about my first question, you'd be looking at a completely different thing if you were in a position to substitute variables, not a differential equation...

 

I understand what the total derivative does, when variables depend on other variables, but - and I'm proposing a totally improvised idea, I haven't tried to do anything with it mathematically yet - why can't you differentiate something with respect to all variables, even when those variables don't rely on each other (just a thought, there's probably some topological reason why )

 

Something like

 

[math]

z = f(w, x, y)

[/math]

 

[math]

\frac{dz}{dwdxdy} = ...?

[/math]

 

looks pretty stupid, but I don't see why it couldn't work...

[mathematic]

hmm, I see what you mean about my first question, you'd be looking at a completely different thing if you were in a position to substitute variables, not a differential equation...

 

I understand what the total derivative does, when variables depend on other variables, but - and I'm proposing a totally improvised idea, I haven't tried to do anything with it mathematically yet - why can't you differentiate something with respect to all variables, even when those variables don't rely on each other (just a thought, there's probably some topological reason why )

 

Something like

 

[math] z = f(w, x, y) [/math]

 

[math] \frac{dz}{dwdxdy} = ...? [/math]

 

looks pretty stupid, but I don't see why it couldn't work...

[math] \frac{d^3z}{dwdxdy} = [/math] is a well defined concept (using ∂ rather than d).

[timo]

Since no one has bothered telling you, yet: The TeX code for a partial derivative symbol is "\partial" (who would have guessed ).

Edited April 4, 2012 by timo