What's the difference between dx and ∂x?

In summary, dx and \partial x mean an infinitesimal change in x, but they're not equal because dx could depend on y, for example. And x could depend on y, for example.
  • #1
izabo
3
0
I understand from the equation:
[tex] df = {\frac{\partial f}{\partial x}} dx + {\frac{\partial f}{\partial y}} dy [/tex]
that: [itex] dx \neq \partial x [/itex].

I understand why [itex] df \neq \partial f [/itex]
([itex]df[/itex] is the change in f when the change in all the variables is infinitesimal, and [itex] \partial f [/itex] is the change in f when the change in one vriable is infinitesimal and the others are constant, right?).

but doesn't both [itex] dx [/itex] and [itex] \partial x [/itex] mean an infinitesimal change in x? if so, why aren't they equal? if not what do they mean?
 
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  • #2
x could depend on y, for example. Or both x and y could depend on some common other variables. In the same way as ##df \neq \partial f##, ##dx \neq \partial x##
 
  • #3
mfb said:
x could depend on y, for example. Or both x and y could depend on some common other variables. In the same way as ##df \neq \partial f##, ##dx \neq \partial x##

how could x depend on y? aren't they both independent variables?
 
  • #5
mfb said:
Who said that?

does it really matter?

btw i meant that for f(x,y) if that wasn't clear.
 
  • #6
izabo said:
[tex] df = {\frac{\partial f}{\partial x}} dx + {\frac{\partial f}{\partial y}} dy [/tex]
that: [itex] dx \neq \partial x [/itex].
If you want to make sense of this for a function ##f:\mathbb R^2\to\mathbb R## (without using definitions from differential geometry), then you should define ##df:\mathbb R^4\to\mathbb R## by
$$df(x,y,z,w)=D_1f(x,y)z+D_2f(x,y)w$$ for all ##x,y,z,w\in\mathbb R##. This ensures that for all ##x,y,dx,dy\in\mathbb R##, we have
$$df(x,y,dx,dy)=D_1f(x,y)dx+D_2f(x,y)dy.$$ What you wrote can be considered a sloppy notation for this result.

This is a way to make sense of df, dx and dy. I don't think I have ever seen anyone try to make sense of ##\partial x##. It's just a small part of the notation ##\partial f/\partial x##, which means ##D_1f##. I find the notation ##\partial f/\partial x## misleading, since it hides the fact that ##D_1f## (the partial derivative of f with respect to the first variable slot) is the same function no matter what symbols we usually use to represent the numbers we plug into ##f##.

Note that with this definition, there's no need for dx and dy to be "infinitesimal". What does that even mean? I know that there's a definition of that term, but I haven't studied it, and I'm pretty sure that almost no one of the authors of physics books who use that term have either. If you see the term "infinitesimal" in a physics book, you should assume that it has nothing to do with infinitesimals. You should interpret it as a code that let's you know that the next thing that follows is a first-order approximation. For example, to say that for infinitesimal x, we have ##e^x=1+x##, is to say (in a weird and confusing way) that there's a function ##R:\mathbb R\to\mathbb R## such that ##e^x=1+x+R(x)## for all ##x\in\mathbb R##, and ##R(x)/x\to x## as ##x\to 0##.

df(x,y,z,w) is a first-order approximation of the difference f(x+z,y+w)-f(x,y). So if we want to be weird and confusing, we can say that for infinitesimal z and w, we have f(x+z,y+w)-f(x,y) = df(x,y,z,w). If we want to cause some additional confusion, we can use the notations dx and dy for the real numbers z and w, and then drop the (x,y,z,w) from df(x,y,z,w), and the (x,y) from ##D_1f(x,y)## and ##D_2f(x,y)##. Then we would be saying that for infinitesimal dx and dy, we have
$$f(x+dx,y+dy)-f(x,y)=df=D_1f dy+ D_2f dx =\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy.$$
 
Last edited:
  • #7
Δ = difference

d = Δ but small difference, infinitesimal

δ = d but along a curve

Mathematical symbols are always graphics.I’m not sure if that will be true, but it would be beautiful.
 

1. What is the meaning of dx and ∂x in mathematical notation?

Dx and ∂x are both symbols used in mathematical notation to represent infinitesimal changes in a variable. However, they have different meanings and uses in various mathematical contexts.

2. What is the difference between dx and ∂x in calculus?

In calculus, dx represents an infinitesimal change in the independent variable, usually denoted as x. It is used in derivatives and integrals to represent the limit as the change in x approaches 0. On the other hand, ∂x represents a partial derivative, which is used to calculate the rate of change of a function with respect to a specific variable while holding all other variables constant.

3. Can dx and ∂x be used interchangeably in mathematical equations?

No, dx and ∂x cannot be used interchangeably in mathematical equations. They have different meanings and cannot be substituted for one another. Using the wrong symbol can lead to incorrect results and misunderstandings in mathematical calculations.

4. Are dx and ∂x used in the same way in all branches of mathematics?

No, the use of dx and ∂x may vary in different branches of mathematics. In calculus, they are used in derivatives and integrals, while in differential geometry, they are used to denote differentials and covariant derivatives. It is important to understand the context in which these symbols are used to avoid confusion.

5. How can I remember the difference between dx and ∂x?

A helpful tip to remember the difference between dx and ∂x is to think of dx as representing a small change in the independent variable, while ∂x represents a partial change in a specific variable. Additionally, understanding the specific context in which these symbols are used can also aid in remembering their meanings.

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