Does dx have mulitple personalites?

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In summary: I don't know what dx actually is supposed to mean behind the integral sign, I'm really confused.Can someone clarify?In summary, dx is a real variable that is supposed to represent an infinitessimally small quantity. It is necessary in the definition of a derivative, and can also be abused in flat spaces to represent a measure.
  • #1
Johnny
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Can someone, who really knows and understands, tell me what dx (or whatever variables given) means behind the integral sign? I have seen more disagreement in Calculus books concerning this. Some authors say it's there just to "indicate the variable" your integrating with respect to, i.e., its not formally required. I had a mathematics professor tell me otherwise, namely that dx is a "real" variable and is required, multiplying at every point with f(x), defining a virtual infinite number of Reimannian rectangles as in S f(x)dx, (S means sum) and performing the normal integration.

Also, (and this ties in) if I have for example, a simple separable ODE such as:

m dv/dt = mg -kv

(where g is accelleration due to gravity, m is mass, v is velocity and k is frictional constant), and if we assume dv/dt is the standard Leibniz operator notation for a derivative, then how can one simply multiply dt through, when it's actually part of an operator? Now I have read that by "appropriately" defining dt, then defining dv as dv = v' dt, we get of course v' = dv/dt. Now, having defined dv/dt as a ratio of differentials, using the same Leibniz notation, we should NOW be able to use basic algebraic techniques with little worry. But wait, if we now integrate both sides:

S (1/(g - cv))dv = S dt where S means sum, and c = k/m,

I still don't know how to interpret S dt? Or now S (1/g-cv)dv for that matter? Do I sound confused? I am! Any help on this is greatly appreciated. I need to understand this.
 
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  • #2
Yes

[itex]dx[/itex] does indeed have "multiple personalities".


The intuition is that [itex]dx[/itex] is "supposed" to be an "infinitessimally small quantity". The notation for many concepts in analysis was specifically chosen so that it looked like you really were manipulating infinitessimally small quantities.



Am I presuming correctly that when you write

(1/g - cv)

You mean

[tex]\frac{1}{g - cv}[/tex]

? If so, you wrote it wrong; you need to wrap the denominator in parentheses, such as: (1 / (g - cv))


Anyways:

[tex]\int \frac{dv}{g - cv}[/tex]

is certainly in the form you mentioned earlier. If we write [itex]f(z) = 1 / (g - cz)[/itex], then we have:

[tex]\int \frac{dv}{g - cv} = \int f(v) \, dv[/tex]

so you can use your favorite intuitive interpretation of [itex]\int f(x)\,dx[/itex] to interpret this integral.
 
  • #3
The entity [itex]dx[/itex] is really something called a "1-form." You can feed a vector field [itex]v[/itex] to a 1-form [itex]\omega[/itex], and it spits out a real-valued function [itex]\omega(v)[/itex].

In more precise terms, a 1-form on a manifold M is a map from [itex]\text{Vect}(M)[/itex] to [itex]C^\infty(M)[/itex] that is linear over [itex]C^\infty(M)[/itex]. [itex]\text{Vect}(M)[/itex] is the set of all vector fields that can be defined on M, and [itex]C^\infty(M)[/itex] is the set of all smooth (real-valued) functions on M.

So, really, the differential [itex]dx[/itex] has only one real personality -- that of a 1-form. Much of this formality is not needed in basic calculus classes, however, and this 1-form can be somewhat abused so that it appears to take on other roles, particularly in flat spaces.

- Warren
 
  • #4
in addition to being a 1-form, i was going to mention that dx is also a measure, which is a totally different kind of gadget.

i would say, yeah, it definitely has multiple personalities.
 
  • #5
Originally posted by lethe
in addition to being a 1-form, i was going to mention that dx is also a measure, which is a totally different kind of gadget.

Yeah, I forgot about that.. hehe... suppose you're right, it's an entity that can function as anyone of a handful of mathematical devices.

- Warren
 
  • #6


Originally posted by Hurkyl
[itex]dx[/itex] does indeed have "multiple personalities".


The intuition is that [itex]dx[/itex] is "supposed" to be an "infinitessimally small quantity". The notation for many concepts in analysis was specifically chosen so that it looked like you really were manipulating infinitessimally small quantities.



Am I presuming correctly that when you write



You mean

[tex]\frac{1}{g - cv}[/tex]

? If so, you wrote it wrong; you need to wrap the denominator in parentheses, such as: (1 / (g - cv))


Anyways:

[tex]\int \frac{dv}{g - cv}[/tex]

is certainly in the form you mentioned earlier. If we write [itex]f(z) = 1 / (g - cz)[/itex], then we have:

[tex]\int \frac{dv}{g - cv} = \int f(v) \, dv[/tex]

so you can use your favorite intuitive interpretation of [itex]\int f(x)\,dx[/itex] to interpret this integral.

Sorry about the parentheses; I corrected it. Actually, I want to know what dx represents behind the integral sign.

Why is it needed, other than to indicate the variable your integrating with respect to? In my calculus book, the first time dx shows up behind the integral, is in the definition of an integral as the limit of a Reimann sum. But my author indicates that dx in that context is not to be confused with dx as it applies to a derivative. I find that hard to believe. But because I'm not a mathematician, for now, I'm going to assume I don't understand. He mentions it's there only for convenience, and useful for such purposes as u du substitution. Once I understand this, I think manipulation of dx in a DE will make sense. In the ODE example, dx is part of a derivative one second, then (by simple multiplcation) behind the integral in the next. Does dx mean the same thing in both cases? I know it's required for a derivative, but what about the integral? What do you think?
 
  • #7
Does this manipulation make it easier to digest?

[tex]
m v'(t) = mg - kv(t)
[/tex]
[tex]
\frac{1}{g - c v(t)} v'(t) = 1
[/tex]
[tex]
\int \frac{1}{g - c v(t)} v'(t) \, dt = \int 1 \, dt
[/tex]

By substitution, we know

[tex]
\int \frac{1}{g - c u} \, du = \int \frac{1}{g - c v(t)} v'(t) \, dt
[/tex]

Which let's us finish off the problem.
 
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1. What is Dx?

Dx is not a commonly used term in scientific language, so it is difficult to provide a definitive answer. It could potentially refer to a medical or psychological diagnosis, a variable in a mathematical or scientific equation, or an abbreviation for a specific field or concept. Without more context, it is impossible to accurately answer this question.

2. Can someone have multiple personalities?

This is a complex question and one that is still debated in the field of psychology. The term "multiple personalities" is not used in modern diagnostic criteria, but it may refer to Dissociative Identity Disorder (DID), a condition in which a person experiences multiple distinct identities or personalities. Some experts believe that DID is a coping mechanism for trauma, while others argue that it is a cultural construct. It is important to note that a proper diagnosis can only be made by a trained mental health professional.

3. What are the symptoms of multiple personalities?

Again, the term "multiple personalities" is not used in modern diagnostic criteria. For DID specifically, symptoms may include experiencing gaps in memory, feeling detached from oneself, and having distinct identities with unique behaviors and attitudes. However, these symptoms can also occur in other mental health disorders, so it is important to seek a professional evaluation for an accurate diagnosis.

4. Is DID a real disorder?

While there is still debate in the field of psychology, DID is recognized as a valid diagnosis in the DSM-5 (Diagnostic and Statistical Manual of Mental Disorders). It is important to note that the diagnosis of DID is not without controversy, and further research is needed to better understand and treat this condition.

5. How is DID treated?

Treatment for DID typically involves a combination of therapy, medication, and support. The primary goal is to help the individual integrate their identities and address any underlying trauma that may have contributed to the development of the disorder. Treatment can be a long and difficult process, but it is possible for individuals with DID to lead fulfilling lives with proper support and treatment.

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