Is dx=0 a Valid Statement?

  • Thread starter Arend Heyting
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In summary, the statement "dx=0" is not correct in the usual sense of a differential in Calculus. It has different interpretations depending on the approach to Calculus being used, but it does not have a numerical value and is not equal to 0.
  • #1
Arend Heyting
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Is dx=0 a correct statement?
 
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  • #2
No, if you mean "dx" here in the usual sense of a differential, it is NOT 0.

There are a number of different ways of interpreting "dx" depending on how you are handling Calculus itself. If you are using "Non-standard Analysis" (Calculus becomes very simple in "Non-standard Analysis" but it requires some very deep logic theory to show that it is valid), then "dx" and all differentials are "infinitesmals". If you use the more "standard" limit approach to Calculus, "dx" is purely a "symbol" that has no meaning by itself but can be connected to the ordinary derivative by "dy= f'(x)dx". In either case, it is not 0 (unless, of course, x itself represents a constant function).

(Hah! I finally beat someone by one minute!)
(Usually, I am the one who posts one or two minutes after another.)
 
  • #3
The statement is meaningless. dx is not a number, but a symbol.
For integrals, it defines the integration variable.
For derivatives, it defines which function is being differentiated with respect to which variable.
 
  • #4
Arend Heyting said:
Is dx=0 a correct statement?

If x is held constant, sure. We need context, are you taking a math class?
 
  • #5
As the general answers say, the statement is rigorously meaningless unless used in very special (and typically advanced) scenarios. On an intuitive infinitesimal level, I'd say it means that ##x## is constant as algebrat says.
 

1. Is dx=0 a true statement?

Yes, dx=0 is a true statement in certain contexts, such as when dealing with a constant function or when using a limit to approach a specific value.

2. Can dx=0 ever be false?

In most mathematical contexts, dx=0 is considered to be a true statement. However, in some situations, such as when dealing with limits of functions with undefined values, it can be considered false.

3. What does dx=0 actually mean?

The notation dx=0 is often used in calculus to represent an infinitely small change in the independent variable x. It can also be interpreted as the derivative of a constant function, which is always equal to 0.

4. Is dx=0 used in other areas of science?

Yes, the notation dx=0 is used in various fields of science, such as physics and engineering, where it is used to represent infinitesimal changes in physical quantities.

5. Are there any exceptions to dx=0 being a true statement?

There are some exceptions to dx=0 being a true statement, such as when dealing with discontinuous functions or when using non-standard analysis which allows for infinitesimal quantities to be non-zero. However, in most mathematical contexts, dx=0 is considered to be a true statement.

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