triclino

yea we will go step by step, just keep asking questions.

 

So this is just vector v, which consists of its magnitude and direction, since v is moving in a circular path the sin and cos show which direction it is pointing:

 

 

For example if it is at the angle [math] \theta = 90 [/math] (at the top) the vector is [math] \mathbf{v} = v (-\hat{\mathbf{x}}) [/math] pointing only left in the x direction.

This is my main objection , how do we prove that v is perpendicular to r

Okay so this is what I was able to do:

 

[math] {\mathbf{r}} = r \hat{r} = r ( cos\theta\hat{\mathbf{x}} + sin\theta\hat{\mathbf{y}}) [/math]

[math] {\mathbf{v}} = v (-sin\theta\hat{\mathbf{x}} + cos\theta\hat{\mathbf{y}}) [/math]

 

[math] \mathbf{a} = \frac{d\mathbf{v}}{dt} = v (-\dot{\theta} cos\theta\hat{\mathbf{x}} - \dot{\theta}sin\theta\hat{\mathbf{y}}) [/math]

 

1) [math] \mathbf{a} = - v\dot{\theta}\hat{\mathbf{r}} [/math]

 

2) [math] \dot{\theta} = \frac{d( \omega t)}{dt} = \omega [/math]

 

[math] \dot{\mathbf{r}} = \mathbf{v} = r \dot{\theta}(-sin\theta\hat{\mathbf{x}} + cos\theta\hat{\mathbf{y}}) [/math]

 

3) [math] |\mathbf{v}| = v = \sqrt{ (r \dot{\theta})^2((-sin\theta\hat{\mathbf{x}} + cos\theta\hat{\mathbf{y}}))^2} = r \dot{\theta} [/math]

 

Then using 1), 2) and 3 you get [math] a = \frac{v^2}{r} [/math]

 

Is this what you were wanting?

Thanks ,but you have to explain each step because i am a bit confused:

How do you get :

[math] {\mathbf{v}} = v (-sin\theta\hat{\mathbf{x}} + cos\theta\hat{\mathbf{y}}) [/math]

It would still be geometry/trig. What is your concern with it?

proof without using geometry at all ,but only vector analysis.

In the proof that i read in most books they use the properties of similar triangles .

In proving that r.r=r^2 you can do that without using the fact that cosθ =1

http://en.wikipedia.org/wiki/Proof_verification

 

http://us.metamath.org/index.html

 

I think you'd enjoy the latter site.

I think you would enjoy the high school problem in post #7.

D.H the word is foundations .

Do you know which are the foundations of mathematics ??

What is your definition of mathematical proof??

Do you perhaps know any efficient way of checking why a mathematical proof is correct or not??

Here is a high school mathematical proof (to start with easy one).

Prove: |x|<y

proof:

|x|<y <=> [math]|x|^2<y^2\Longleftrightarrow x^2-y^2 <0[/math] <=> (x+y)(x-y)<0 <=> {[(x-y)<0 and (x+y)>0] or [(x-y)>0 and (x+y)<0]}.

IF [(x-y)<0 and (x+y)>0] <=> x<y and x>-y <=> -y<x<y.................1

IF [(x-y)>0 and (x+y)<0] <=> x>y and x<-y...................................2

From (1) and (2) we conclude that : |x|<y <=> -y<x<y

Is that proof correct or not???

In most physics books ( even the University ones) i looked at, they use geometry to derive that the acceleration is [math]\frac{v^2}{r}[/math],where v is the constant speed and r is the radius (magnittute of the position vector r) of the motion.

Can we not use pure vector analysis to derive the above fact??

Why ,when x=2a and y=2b ,then x+y=2a+2b?.What is the axiom ,theorem or definition supporting that statement??.

From how many arguments your proof consists of??

Can you isolate each argument??.

For example can you say :

Argument No 1 : state the argument

Argument No 2: state the argument

Argument No 3 : state the argument .............e.t.c........e.t.c

Merged post follows:

I mean logical very loosely here, not in relation to mathematical logic really.

 

Your idea of a proof seems correct to me. Going from one starting statements/axioms etc to another statement "the theorem/axiom" etc. in steps that cannot be disputed.

 

A simple example (can be found on Wikipedia in fact.)

 

Theorem The sum of two even integers is itself even.

 

(This is our statement we wish to prove, i.e. end up with this as an infallible truth.)

 

Proof

Let [math]x,y[/math] be two even integers. Since they are even, they can be written as

 

[math]x=2a[/math] and [math] y=2b[/math]

 

for integers [math]a[/math] and [math]b[/math].

 

(First question about rigour and standards of proof. Is the above statement obvious? Can I take it as "given" or should I prove it viz a lemma?)

 

Yes this the right question ,why x=2a and y=2b ,what is the axiom ,theorem or definition supporting those statements.

The answer here is the following:

Definition:

for all ,x : x is even => there exists integer ,a such that x=2a

Or in quantifier notation :

[math]\forall x[/math][ x is even[math]\Longrightarrow\exists a( a\in Z\wedge x=2a)][/math].

The final statement x=2a ,y=2b ,where a,b are integers is concluded in the following way:

step 1 : Assume ,x,y to be even

step 2 : bring in the above definition of even Nos:

[math]\forall x[/math][ x is even[math]\Longrightarrow\exists a( a\in Z\wedge x=2a)][/math]

step 3: What do you think step 3 should be???

Here is an alternative proof:

Let A be the area of triangle

x ,the height

z the side of the triangle

y the base of the triangle

And , θ the angle between the height ,x and the side z

Now all the above except , θ ,change with time . We also know dz/dt .

So if you can express A in term of ,θ and z ,then the calculation of dA/dt will be an easy matter.

So,the area of the triangle is:

A = (y.x)/2.........................................................................................1

But :

sinθ = [math]\frac{\frac{y}{2}}{z} = \frac{y}{2z}\Longrightarrow[/math] 2zsinθ = y.........................................................................................2

cosθ = [math]\frac{x}{z}\Longrightarrow[/math] zcosθ = x......................................................................................................3

And substituting (2) and (3) into (1) we have:

A= sinθcosθ[math] z^2[/math]

Now we can differentiate w.r.t time ,t.

Thus :

[math]\frac{dA}{dt}[/math] = sinθcosθ[math]\frac{d(z^2)}{dt}[/math]=2zsinθcosθ[math]\frac{dz}{dt}[/math],but since dz/dt = 2ft/min and zcosθ =x we have that:

[math]\frac{dA}{dt}[/math]= 4xsinθ (ft\min). And since we want the rate when x=8ft ,then we have:

[math]\frac{dA}{dt}[/math] = 32sinθ ([math]\frac{ft^2}{min}[/math]).

Where θ =arctan[math]\frac{\frac{19}{2}}{8}[/math],if when x=8ft ,y=19ft

Hello, this is my first post. I have a question of related rates that was giving me trouble, could someone help me, it is...

 

There is an isosceles triangle with the sides increasing at a rate of 2 ft/min, the base is 19 feet, what is the rate of change of area when the height is 8 ft?

 

Someone said to me take the derivative with respect to a variable other than t, this did not make sense to me because you want rate of growth per unit of time.

 

 

Tacobell

Tacobell

The only thing that does change with time is the angle, θ between the side of the triangle and its height.

So if you can express the area of the triangle in terms of this angle and one of its sides ,then very easily you will find the:

dA/dt of the triangle,because you know the d(one side of the triangle)/dt = 2ft/min

 

And, we have the theorm "If a real-valued function f(x) is continuous on (0,infinity) and its limit, as x tends to infinity, exists (and is finite), then f(x) is uniformly continuous."

Can you find anywhere such a theorem?

But, you shouldn't go the first step directly. You must use Left and Right limits (for both x and y).

Now if this is not another pack of mathematical nonsenses what is it then?

You keep on writing non existing theorems and conclusions ,by simply imagining them.

"Compact interval" is not involved here. It is a separate point.

Yes you can escape by writing nonsenses because the moderators of this forum

perhaps are not capable of detecting such unfounded mathematical nonsenses that you keep on writing.

Check the following: http://en.wikipedia.org/wiki/Uniform%5Fcontinuity

Go to the last point in "Properties" section.

I see you have a problem in reading :

The section under properties it says that:

if the interval on which A continuous function f is defined is COMPACT then the function is uniformly continuous.

Is the interval (0,[math]\infty[/math]) on which f is defined COMPACT???

It uses the definition of a Bounded Sequence.

So, how does the definition of a bounded sequence help to prove that the sequence has a limit .

Read the question again ,it says to prove that the sequence has a limit by using the theorem (axiom) of nested intervals

1. You will have to check the Left Limit for the variable x. It will be lim f(x,y)=xo*y

2. Check the Right Limit for the variable x. It will be also lim f(x,y)=xo*y.

3. Therefore, the Left Limit of f(x,y), as x tends to xo, is equal to the Right Limit.

4. Repeat the previous, as y tends yo. And, you will get the same results.

5. Then lim f(x,y), as (x,y) tends to (xo,yo), is equal to f(xo,yo).

 

Note: There are some simple functions that can be considered continuous, unless it is required to prove so....... such as f(x)=x, f(x)=x^2.

You mean that:

IF [math]lim_{y\to y_{o}}(lim_{x\to x_{o}}f(x,y))=f(x_{o},y_{o})=x_{o}.y_{o}[/math]..................... THEN

[math] lim_{(x.y)\to(x_{o},y_{o})}f(x,y) =f(x_{o},y_{o})=x_{o}.y_{o}[/math]???

Since the sequence is bounded from above, therefore it reaches a certain number (Ao), as n tends to infinity. Then, lim A(n), as n tends infinity, equals Ao. Therefore, it has a limit.

On what theorem or axiom you base such a conclusion.

Sorry, another factor is remaining which is lim f(x), as x tends to infinity, exists and is equal to zero.

 

And, we have the theorm "If a real-valued function f(x) is continuous on (0,infinity) and its limit, as x tends to infinity, exists (and is finite), then f(x) is uniformly continuous."

It seams to me that you do not know the definition of uniform continuity.

As for the theorem you stated it exists nowhere in the whole of cosmos.

Anyway if you still insist on the validity of your theorem ,then give a proof.

O.K...... I was talking like that because f(x,y) is obviously continuous,

and lim f(x,y), as (x,y) tends to (xo,yo), is f(xo,yo).

If you need an exact proof, then find the left limit which will be f(xo - epsilon , yo - epsilon). And, find the right limit which will be f(xo + epsilon , yo + epsilon). Then, since both are equal then the limit is found.

How can you prove that left and right limits are equal??

How do you know that f(x,y) is continuous at ([math]x_{o},y_{o}[/math]),without proving it

That a function to be continuous on an interval, it has to be continuous at every point in the interval.

That is the definition of a function being continuous on an interval and not the definition of a function being uniformly continuous over an interval

The only discontinuity is at x=-1, which is not included in the domain (0,infinity). Therefore the function is uniformly continuous in(0,infinity).

Your conclusion is based on what definition or theorem?

A proof is a series of logical arguments or steps from one collection of statements to another.

 

Can you give an example to justify your definition?

Hence to produce a proof consisting of "logical arguments" where those arguments can be isolated and examined for there validity in an indisputable and clear way??

By the way in logic we do not have "logical arguments" ,but valid or not valid arguments

I think every mathematical forum at the very beginning of its existence should produce a definition of the mathematical proof and then according to that definition every mathematical proof should be written down.

Also a check process should be mention ,hence to avoid useless duals between the members of the forum whether a proof is correct or not.

I think the above should be stipulated as the basic rules of each and every mathematical forum

What is meant by "monotone sequence"?

Is a sequence increasing or decreasing

What Euclidean Norm and Maxnorm have to do here? You can easily get the limit by substituting with x=1 and y=1, you will get lim f(x,y)=1*1=1.

That is not a proof is there??

How do we -prove that:

The axiom (theorem ) of the sequence of the nested interval in real Nos implies that every monotone sequence in real nos ,bounded from above, has a limit