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triclino

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Posts posted by triclino

  1. The product rule:

     

    [math]

    y=u(x)\cdot v(x)

    [/math]

    [math]

    y = u\cdot v

    [/math]

    [math]

    y+dy=(u+du)\cdot(v+dv)

    [/math]

    [math]

    y+dy=uv+udv+vdu+dudv

    [/math]

    now [math]dudv[/math] is discarded on the grounds of being "too small".

    If I were to include it, later on (by subtracting y and dividing through by dx), it would become [math]\frac{dudv}{dx}[/math]. What does that mean?

     

    Definition: if y = f(x) ,then dy= f'(x)Δx ,and dx = 1.Δx.

     

    Theorem : if y = u(x)v(x) ,then dy = duv + udv.

     

     

    hence y+dy = vdu +udv +uv.

     

    Thus [math]y+dy\neq(u+du)(v+dv)[/math].

     

    However y+Δy = u(x+Δx)v(x+Δx) = (u+Δu)(v+Δv)

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

     

     

    So [math]\neg(0<-1)[/math] and 1>0 are not true since the arguments are invalid??

     

    Besides [math]\neg(0<-1)[/math] and 1>0 not only are true ,but can be logically concluded in the following way:

     

    2>1 => 2-1>1-1 => 1>0

     

    [math]\neg(1<0)\Longrightarrow 1\geq 0\Longrightarrow 1-1\geq -1\Longrightarrow 0\geq -1\Longrightarrow\neg(0<-1)[/math]

  3. If we are presented with the following arguments :

     

    1) If 1<0 ,then 1-1< -1 => 0<-1. But since [math]\neg(1<0)[/math] ,then [math]\neg(0<-1)[/math].

     

    2) if 1>0 then 1+1>0+1 => 2>1 . And since 2>1 ,then 1>0

     

    What true facts can we use so that we can decide whether the above arguments are valid or non valid??

  4. Yes it is but in the book the 'm' is missing from the denominator so I am assuming that there is a typo in the book as without the 'm' being in the denominator the answer cannot be 1 - a/g.

     

    If there was a mechanism to display an image of the page from the book then I think it would be easier to show what the problem is.

     

     

    Yes there is a typo in the book. No doubt about it

  5. The question is asking for a ratio of force to a persons weight.

     

    The solution in the book states the following:-

     

    F(rope) = mg - ma and the weight of person = mg

     

    therefore the ratio of F(rope) to their weight = (mg - ma)/g = 1.0 - a/g

     

    So my question was how do you get from (mg - ma)/g to 1.0 - a/g

     

    The ratio of F(force) to their weight is :

     

    (mg-ma)/mg = 1-a/g

  6. Can the following two definitions of sqrt(x) be considered as equivalent??

     

     

    1) if [math]x\geq 0[/math] ,then ([math]\sqrt x=y[/math]) if ([math]y^2=x\wedge y\geq 0[/math])

     

     

    2) if [math]x\geq 0[/math] ,then ([math]\sqrt x=y[/math]) iff ([math]y^2=x\wedge y\geq 0[/math])

  7. But ,however, logic dictates us the following:

     

    If we put :

     

    ([math]x\geq 0[/math]): = p

     

     

    (|x|=x) := r

     

    (x<0): = q

     

     

    (|x|= -x): = s...............then we have:

     

    ([math]p\Longrightarrow r[/math]) and ([math]q\Longrightarrow s[/math]) which logicaly implies:

     

    [math]p\wedge q\Longrightarrow r\wedge s[/math]

     

    But p&q means that we have : [math]x\geq 0[/math] and x<0 .

     

    IS that possible??

  8. Suppose I have a small (infinitesimal) quantity [math]dy[/math] and another small quantity [math]dx[/math] and they are related by [math]dy = k \cdot dx[/math]. Will that automatically imply that [math]\frac{dy}{dx}=k[/math] is the derivative [math]\left(\frac{\mathrm{d}y}{\mathrm{d}x}=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)- f(x)}{\Delta x}\right)[/math] of [math]x[/math] with respect to [math]y[/math]?

     

    I have seen several examples of such things occuring in engineering textbooks, such as electrical relations between the charge on a capacitor and the voltage across it. (I can't remember the details, and my notes are safely stored in the basement).

     

    Given y = f(x) ,then we define : dy = f'(x)Δx and Δx= 1.dx .

     

    Hence by definition: dy/dx = f'(x) ,where dy/dx is the ratio of the differentials dy ,dx

     

    In the case where the derivative is denoted by :[math]\frac{dy}{dx}[/math],then this is equal with the ratio of the two differentials dy/dx

  9. No, this is standard notation. The floor function is [math]\lfloor x\rfloor[/math].

     

     

     

    Plug it into the definition of uniform continuity and check that the definition is satisfied.

     

    You mixing up uniform continuity with uniform convergence .

     

    But in my very 1st post i ask if any body could prove uniform convergence that i could not prove.

     

    Now by producing a No and plugging it into the definition of uniform convergence to find out if the definition is satisfied or not ,i am sorry is not much of a help

  10. O.k this a trivial proof and i am waiting for another trivial proof :

     

    Suppose x does not belong to the set ,A .But x does not belong to the empty set.

     

    Hence if x does not belong to A ,then x does not belong to the empty set .

     

    Thus : [math]\emptyset\subseteq A[/math]

  11. Learn your symbols, triclino. [math]\lceil x\rceil[/math] is the the ceiling function. For example, [math]\lceil \pi \rceil = 4[/math].

     

    Edit

    I should have said [math]N=\lceil 1/(4\epsilon^2)\rceil+1[/math] to handle the rare case where [math]1/(4\epsilon^2)[/math] is an integer.

     

    You mixing up ceiling function with the floor function.

     

    But according to what axiom or theorem you came to the conclusion :

     

    [math]N=\lceil 1/(4\epsilon^2)\rceil+1[/math] or [math]N=\lceil 1/(4\epsilon^2)\rceil[/math]

  12. .

     

    This doesn't make a lick of sense: [math]\vec a \cdot \vec b \cdot \vec c[/math]. That can only mean triclino was talking about the cross product, and what he wrote isn't correct for that either.

     

     

    Why you did not ask me what i meant ,but make such a fuss over minor details ??

     

    This is a physics forum and people know what a dot product is , and very easily can understand that:

     

    A.(B.C) is really A(B.C) since the dot product is always a scalar.

     

    Now is not true that if the vectors are on the same currier or parallel then :

     

    A(B.C) =(A.B)C ???

  13. Let : [math]f_{n}(x) =\frac{x}{x^2+n}[/math] be a sequence of functions in the Real Nos.

     

    I can prove point wise convergence to the zero function as n goes to infinity .

     

    But is there a uniform convergence ??

     

    If there is, can anyone prove it ,please??

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