Difference Between Partial and Ordinary Differentials

[Pinu7]

I have been wanting to ask this for a while.

In Calc I, I was introduced to differentials. It seemed like they act like quantities(please corrected me if I'm wrong). For example dx/dx=1. You can obtain this by differentiating x or by eliminating the dx in the numerator and denominator(I do not know why this worked).

What convinced me that differentials where quantities was the chain rule. dy/dx=(dy/du)(du/dx). The proof is a bit tough, but you will obtain the same result by eliminating the du.(I may be making a TREMENDOUS mathematical blunder here, but it seemes to work)

In Calc III, I was introduced to $$\partial$$x and$$\partial$$y. Obviously I found out that $$\partial$$x$$\neq$$dx or else the chain rule for multiple variables would not simplify to dz/du.

So, why are these two infinitesimals so different?

 

[HallsofIvy]

No, you were not "introduced to $\partial x$ and $\partial y$ in Calc III. You were introduced to the partial derivatives $\partial f/\partial x$ and $\partial f/\partial y$. There is no such thing as a "$\partial x$".

One important reason is that the partial derivatives themselves just don't tell you enough about the function. If the derivative of a function of one variable exists at a point, then it is differentiable (and so continuous) at that point. A function of several variables can have all its partial derivatives at a point and still not be differentiable nor even continuous at that point.

Take f(x,y)= 0 if xy= 0, 1 otherwise. It is easy to show that $\partial f/\partial x= \partial f/\partial y= 0$ at (0,0) but f is not even continuous there.

 

[Pinu7]

Thanks, that cleared things up for me, HallsofIvy.