Differential Operator

1. Dec 13, 2005

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.

2. Dec 13, 2005

incognitO

think of it....

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

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

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

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

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

EDIT:

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

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

Last edited: Dec 13, 2005
3. Dec 13, 2005

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

4. Dec 13, 2005

incognitO

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

EDIT:

Not to mention Calculus of Variations.

5. Dec 13, 2005

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

6. Dec 13, 2005

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 $$f(x,y)=x+y$$?

7. Dec 13, 2005

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.

8. Dec 13, 2005

Fascinating.

9. Dec 14, 2005

matt grime

Here's another reason for the disticntion.

I give you y, jsut 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.

Last edited: Dec 14, 2005