Total derivative formula nonsense

[mathboy]

Total derivative formula confusion

It took me over an hour to fully resolve the confusion that appears in textbooks about the total derivate formula. Some textbooks use the term total derivative if a function f is a function t and other variables, and each of those variables themselves are functions of t. I'm going to challenge the formulas they give, including the first formula in this wikipedia page:
http://en.wikipedia.org/wiki/Total_derivative

Here is my refutation. Please tell me if I'm right or not.

 

[malawi_glenn]

What has this to do with homework?
The total derivative just follows from the chain rule:

$$\dfrac{d}{dt} \Phi (t,x(t),y(t),z(t)) = \dfrac{\partial \Phi}{\partial t} \dfrac{d(t)}{dt} + \dfrac{\partial \Phi}{\partial x} \dfrac{dx}{dt} + \dfrac{\partial \Phi}{\partial y} \dfrac{dy}{dt} + \dfrac{\partial \Phi}{\partial z} \dfrac{dz}{dt}$$

 

[HallsofIvy]

I don't really see a "refutation" in that. The three definitions of "total differential" given in the Wikipedia article you cite are just three slightly different ways of phrasing the same definition.

 

[malawi_glenn]

abuse and abuse, perhaps a bit sloppy, you'll the other things elsewere, this one for instance:

http://mathworld.wolfram.com/ChainRule.html

 

[mathboy]

The textbooks should not teach with sloppy notations. It made me lose an hour of my homework time resolving something that shouldn't have been there in the first place (putting the exact same f on both sides of the equation).

 

[malawi_glenn]

How many textbooks have you constulted? And what kind? How could you spend one hour resolving that?

 

[CPL.Luke]

the f should appear on both sides of th equation as your differentiating f, the partial and the derivative are two very different things.