triclino

If f(x) = 1 if x is rational and f(x) =0 if x is irrational ,then

f(x)dx is the differential of which function??

Is that an exact or inexact?

You do not seem to have read my original post carefully .We are not examining

[math]W = \int_C \vec F \cdot d\vec l[/math] ,but the scalar product :[math]\int Fdl[/math] as the OP points out.

But let me ask you a question:

Suppose we are given the differential f(x)dx. How do we know whether this differential is excact or inexact ??

from mathematics we have :

W=[math]\int F.dl[/math] => [math]\frac{dw}{dl} = F[/math] => dw = [math]\frac{dw}{dl}.dl[/math] => dw = F.dl.

Isn't dw an exact differential and thus F.dl ,since dw/dl = F??

Sorry it was a typo i meant to say that δW is an inexact differential.

I corrected my post

[math]\delta W = dl \cdot F[/math]

 

is an inexact differential.

F.dl is an exact differential . If you consider δW as an inexact differential (which you may) ,then we have an inexact differential equal to an exact differential.

And again my problem is how do we mathematically prove that :

if [math] W =\int F.dl[/math] then δW = F.dl

Charlatan ,is Table mountain still there??

In a certain book i read and i quote :

"As in mechanics this work is defined by the integral.

W = [math]\int F.dl[/math].

where F is the component of the force acting in the direction of the displacement dl.In a differential form this equation is written.

δW = F.dl.

where δW represents a differential quantity of work"

Is that correct??

Short answer: Because that is the only value that makes sense.

 

So the sqrt(4) = 2 ,because this the only value that makes sense?

This is not the subject of our discussion now. We discussed that in another thread ,let it rest.

"If the moon is made of cheese, then I am a superman" is valid.

.

First of all this is not an argument and thus cannot be valid or invalid .It is simply an implication and can only be true or false.

It is true because we have false implying false

Merged post follows:

No. You seem to keep having this same problem. I suggest reading up a little on logical fallacies.

I simply asked

Remember in this forum we have not come up with a theorem or axiom or definition deciding for the validity or invalidity of an argument

The argument you can make, however is this: if the moon is made of green cheese, then I'm superman. I am not superman. Therefore, the moon is not made of green cheese. The other way, the argument is invalid and unsound.

 

For example, try this:

If something is a horse, then it has four legs.

My table has four legs.

Therefore, my table is a horse.

 

Is that valid? It is the same as the argument you keep getting confused by, with a specific example where the answer turns out not to be true.

As i said examples do not make up a theory

There's two forms of implication. One is (if p then q) <==> (q or not p). For example "if the moon is made of green cheese, then I'm superman" which would be true. Here, p and q may be anything.

 

Another is the sort used frequently in common speech, where to the above one would ask, "So how does the moon being made of green cheese make you superman?" rather than "Of course that will always be true, since in fact the moon is not made of green cheese". This one is harder to define. I think that what it comes down to is that you can logically deduce that if p then q. This would, however, be true only of very specific p and q. Example: "If a number is even, then it is a multiple of 2".

So the argument : if the moon is made of cheese then i am a superman.But since the moon is not made of cheese ,hence i am not a superman .

Is valid??

In defining implication we say that:

p => q is false, if p is true and q false, and is true for all other cases.

The question is if the definition could be other wised.

For example instead of F =>T defined as T ,why not defined as F

So how would you go about defining differentiation where the is no division?

I don't know what do you mean when you say: "there is no division" ,but since

dy = f'(x)dx ,if you divide by dx ,then dy/dx = f'(x) = [math]\frac{dy}{dx}[/math].

Hence the ratio of dy to dx is equal to the derivative of y w.r.t x

Funny how often people who disagree with everyone tend to think that way about all the people they disagree with. "Everyone else is wrong, only I know the truth..."

 

You do not disagree with me ,but with the book of logic.

If you do not want to learn the difference between implication and logical implication,that is not my problem.

Does not your book of logic have a section explaining logical implication??

M.ponens and M. tollens has nothing to do whether our arguments are valid or non valid.

If an argument does not fit the general form of another valid argument that does not mean that it is invalid

Does not your book of logic have a theorem deciding whether an argument is valid or non valid??

Have a look at the Wikipedia article Nilpotent.

 

If the order is not specified then we usually mean order two, as is usually meant in the physics literature.

That definition does justify the equation:

dy = (x+dx)^2 ,but only the equation : [math](x+dx)^2 = x^2+2xdx[/math]

So here we take differentials to be nilpotent. Thus

 

[math](x+dx)^{2} = x^{2} + xdx + dx x + dx^{2} = x^{2}+ 2 x dx[/math].

What is the exact definition in mathematics of nilpotent ??

I assume that by changing x I change y since y=f(x).

So by changing x by dx, I change y by dy.

My goal is then to find out how much the change in y (dy) is in response to a change in x (dx).

In mathematics every statement equation e.t.c is justified by an axiom ,theorem,definition,or a law of logic ,otherwise mathematics would not be mathematics , but a composition of ideas resulting in a chaotic situation.

Yes a change in x will respectively produce a change in y since y =y(x).

Now since by by definition ,dy = ([math]x^2[/math])'dx = 2xdx ,then y + dy = [math]x^2 + 2xdx[/math] and not equal to [math](x+dx)^2[/math].

Unless you have a different definition for differentials .

Further more there is a difference between infinitesimals and differentials

triclino, here is the truth table for logical implication:

 

[math]

\begin{array}{ccc}

p & q & p\rightarrow q \\

F & F & T \\

F & T & T \\

T & F & F \\

T & T & T

\end{array}[/math]

 

Note that p=>q is true regardless of the value of q when p is false. Not that you can do anything with it, though. If p then q doesn't say a thing about q when p is false, and that is what you did wrong in the original post.

 

Aside: If p then q says quite a bit about p when q is false.

This the truth table for the definition for implication and not for logical implication.

There is a great difference between implication and logical implication.

Read your logic book again carefully.

In a simple implication the value of p =>q depends on the values (T,F) of p and q as the above table shows .

In a logical implication the value of P=>Q is always true irrespective of the values of P and Q.

Merged post follows:

I think what you will find is that you cannot use the axioms and theorems of logic to prove any argument of the form

 

1. If P, then Q.
   
2. Q.
   
3. Therefore, P.
   

 

You cannot prove #3 with the theorems of formal logic. That is why affirming the consequent is a fallacy.

oh ,yes you can

You can prove whether an arguments is valid or non valid thru the axioms or theorems of logic, otherwise we would have a situation similar to the one in this thread.

Merged post follows:

The definition of "valid" suffices for that. The conclusion does not follow from the premises, ergo the argument is invalid. It's like if you asked my by what axiom or theorem I prove that a nonhuman is not human.

Little you know about logic.

 

 

There might be other cases that we do not know where although we deny the antecedent or affirm the consequent are valid

 

 

 

No, never.

Let logic decide that with its axioms and theorems , non of which you have cited so far

Here is an example where we deny the antecedent and the argument is still valid :

we say : If i was rich i would have bought Kennedy Airport ,but since i am not rich i cannot bye Kennedy Airport.

Is it not that argument valid??

There might be other cases that we do not know where although we deny the antecedent or affirm the consequent are valid

O.K then ,why when denying the antecedent or affirming the consequent the conclusions of an argument do not logically follow??

But surely there are no terms in the Taylor expansion that are not there when taking the derivative. I mean, the Taylor expansion must yield zero for the second term, right?

 

If the Taylor expansion is used on [math]y=u(x)\cdot v(x)[/math] would the term [math]\frac{dudv}{dx}[/math] also appear?

 

On another note. Is the differential notation not perfectly valid and equivalent to the difference quotient notation?

If

[math]

\frac{dy}{dx} = \lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}

[/math] then what is [math]\frac{dy}{dx}[/math] in the differential notation?

 

If:

[math]

y = x^2

[/math]

 

Changing x by the amount dx gives the change dy in y:

 

[math]

y + dy = (x + dx)^2

[/math]

 

[math]

y + dy = x^2 + 2x\cdot dx + (dx)^2

[/math]

 

[math]

dy = 2x\cdot dx + (dx)^2

[/math]

 

[math]

\frac{dy}{dx} = 2x + dx

[/math]

 

Suggesting that the rate of change in y with respect to x also depends on dx.

Now you have to argue that dx is negligible, which is handled by limits in the difference quotient way of writing this.

So is the differential notation inferior?

hobz ,how do you know that:

[math]

y + dy = (x + dx)^2

[/math]

Well (1) is denying the antecedent (2) is affirming the consequent so from inspection they're invalid.

Why when "denying the antecedent"or when "affirming the consequent" as you say the arguments are invalid?

When is an argument invalid ??

1) An example or examples cannot support a general statement .

2) In my post #3 i showed that the conclusions of both arguments are logically implied