Why are there two different ops for normal and partial derivatives?

[Treadstone 71]

This may seem like an odd question, but why are there two different ops for the normal and partial derivatives? i.e., $$\frac{d}{dx}$$ and $$\frac{\partial}{\partial x}$$? I don't see a difference if only one is used, since we are always differentiating wrt a single variable anyway.

 

[incognitO]

think of it...

in classical mechanics, you can have functions like the action $L$ which is defined in the form

$$L=L(x,y,z,t)$$,

but $x=x(t)$, $y=y(t)$, $z=z(t)$, so

$$\frac{d L}{d t}=\frac {\partial L}{\partial x} \dot{x}(t)+\frac {\partial L}{\partial y} \dot{y}(t)+\frac {\partial L}{\partial z} \dot{z}(t)$$

which clearly is different from $\partial L/\partial t$.


EDIT:

Sorry, my mistake... the derivative is missing one term. It should be read

$$\frac{d L}{d t}=\frac {\partial L}{\partial x} \dot{x}(t)+\frac {\partial L}{\partial y} \dot{y}(t)+\frac {\partial L}{\partial z} \dot{z}(t)+\frac{\partial L}{\partial t}$$

 

[Treadstone 71]

Interesting. I never encountered those. Then again, I'm not in physics.

 

[incognitO]

The above are functions present in Hamiltonian systems, which are a big subject of study for mathematitians too... Specially in P.D.E.

EDIT:

Not to mention Calculus of Variations.

 

[matt grime]

one came first, d/dx, and the other is a generalization of it, but asking what d/dx of some object is is strictly different from asking what partial d by dx of it is since the former assumes that the other variables (if there are any) are a function of x too. That is to say that if f(x)=x+y then

$$\frac{\partial f}{\partial x}$$

makes sense but

$$\frac{df}{dx}$$

doesn't

 

[Treadstone 71]

Wait, if $$f(x)=x+y$$, wouldn't $$\frac{df}{dx}$$ make sense since y is a constant? That is, $$\frac{df}{dx}$$ does not make sense if it was $$f(x,y)=x+y$$?

 

[matt grime]

And what if y weren't a constant? come on, put the pieces together, you should be able to correct the obvious mistakes that people make! Dear God.

 

[Treadstone 71]

Fascinating.

 

[matt grime]

Here's another reason for the disticntion.

I give you y, just y, now differentiate it with respect to x. What's the answer? dy/dx or 0?

I suppose it is unfair of me to expect you to recognize silly errors from catastrpohically bad ones, not to mention hypocritical perhaps.