a query

[blue89]

hi;

 

 

Let , a ≤ x ≤ b , y₁(x) ≤ y ≤ y₂(x)

,Let ,V be an interval/region ;

a ≤ x ≤ b y₁(x) ≤ y ≤ y₂(x) , z₁(x,y) ≤ z ≤ <z₂(x,y)

 

 

in this case ; might we calculate this integral if we would like to choose our "R" function as any type of light or magnetic wave’s equation ?

 

 

 

_a∫^b y₂(x)∫^y₂(x) z₁(x,y)∫^z₂(x,y) ∇.R.dV

 

 

thanks

Edited September 2, 2016 by blue89

[studiot]

Please confirm the following with reference to my diagrams attached.

 

Sorry for the quality of the hasty sketches.

 

1)

Mark off an interval, ab, along the x axis shown by lines through a and b parallel to the y axis.

Draw in y1(x) and y2(x) as the bounds in the xy plane.

This defines a surface (area) in the xy plane.

 

2) Erect a z axis perpendicular to the xy plane showing the bounding area we are working in.

I have shown this hatched.

 

3) Erect rectangular columns over this area from xy plane to the bonding surfaces given by z1(xy) and z2(xy)

 

This defines the volume we are working in now for your function R

 

The general wave equation is a connection between time and space which introduces an extra variable we have not catered for (time)

To take time into account we have to account for the flux of R crossing the volume boundary as defined above, as well as the waves already within the volume.

 

A more restrictive wave equation is the time independent equation of standing waves.

If the waves are standing there is no flux across the boundaries so time may be discounted and a simple spatial volume integral employed.

 

 

[blue89]

Please confirm the following with reference to my diagrams attached.

 

Sorry for the quality of the hasty sketches.

 

1)

Mark off an interval, ab, along the x axis shown by lines through a and b parallel to the y axis.

Draw in y1(x) and y2(x) as the bounds in the xy plane.

This defines a surface (area) in the xy plane.

 

2) Erect a z axis perpendicular to the xy plane showing the bounding area we are working in.

I have shown this hatched.

 

3) Erect rectangular columns over this area from xy plane to the bonding surfaces given by z1(xy) and z2(xy)

 

This defines the volume we are working in now for your function R

 

The general wave equation is a connection between time and space which introduces an extra variable we have not catered for (time)

To take time into account we have to account for the flux of R crossing the volume boundary as defined above, as well as the waves already within the volume.

 

A more restrictive wave equation is the time independent equation of standing waves.

If the waves are standing there is no flux across the boundaries so time may be discounted and a simple spatial volume integral employed.

 

volint1.jpg

 

 

Welcome studiot. sorry for my delayed reply.

and thanks for your comment.

 

as I see a very good understanding about green - stokes - divergence theorems.

Okay I appreciate it.

 

Really I am unable to dominate this three theorem's well now (probably these three theorems are used in engineering commonly and broadly) . (I mean ,I do not know whether this question might make me reach my result which I am loking for)

The more clearly ,I would ask that whether we may use any of wave's equation as our R function. (like this x² + y² =r² ,z=z (note ,dim=3 there (z=z) )

?? : I am unsure whether these days we are able to write any such wave's (magnetic-voice waves') curve equation as given in cartesian form above.

 

Note : I am not trying to make any relation with time and Dimension now. of course if we be able to find >3D ,we might have option to interpret about time character.

but This will already be one of advanced goal (intellectual subject).

so , you may think there the time parameter as static. (no usage in any equation for this thread)

(I see ,you haven't missd this at your last two sentences. ok , think please it as only geometric standing.) (as flux of t is interrupted,only geometric approach now)

[studiot]

Yes the theorems variously attributed to Gauss, Green, Stokes are much used in Engineering and Physics to generate working equations.

Green's other theorems are also used to help with some integration.

They are also used in applied maths (numerical methods) to convert finite element to boundary element methods.

Normally we do not used such a generally shaped volume of integration

(It must be the mathematician in you that wants to be so general)

because it is too difficult.

The volume is generally a box, cylinder or sphere so we can use appropriate coordinates to simplify the calculation and is called the control volume.

The control volume may be infinitesimal ie dx, dy, dz

or it may be large but finite eg a reservoir.

 

As to time independent functions you can integrate over the region in question potential functions are good.

 

(Is R the best variable to use to describe your function ?)

 

Scalar potentials - eg potential energy are much used in thermodynamics.

Vector potentials can demonstrate static fields.

 

So you need to expand your description of what you want to do.

 

In general you either integrate a variable which is a function of position over some region to obtain the total amount of that variable (eg potential energy) in that region

 

That is

 

[math]\int {U\left( {{x_1},{x_2},{x_3}} \right)} d{x_1}d{x_2}d{x_3}[/math]

Which gets you a sum total.

But this is not integrating an equations, which you also mentioned.

For this you (may) need Greens functions or Fourier transforms etc.

For suitable equations that define functions this will get you another equation defining another function.

A good example of this is integrating slopes, loads and moments on beams

[blue89]

hi,

 

I deteceted that I had written equation (in function form) incorrectly.

 

the correct one should be

 

x² + y² = r ² ,z = c ( c є R or c є [a,b] (if we choose c = z ,then this will define calender )

 

ok. I do not want to use best variable ,why if we would like to use a wave's whole equation , then this will be impossible.

 

I am sharing some photos.

 

 

Rally ,I have written in parametric forms but no change again. (I as whether we might choose our R function just one of these equations above in picture)

 

 

yes, this one shows general usage of Divergence theore in math.

 

 

wow , I should thank to @Klaynos

 

it is exciting to use photos.

I will be able to solve some questions (integral , functional..etc) without using symbols :-)

 

 

But this is not integrating an equations, which you also mentioned.

 

ok. I am writing equal expression via changing it so:

 

f(x,y) = x² + y² , while z = c and assume please c є [a,b]

?? SENSITIVITY (A RESEARCH SUBJECT)

 

I am not a physcist and as Nature - Physics mentioned ,I accept that I have not advanced understanding at basic physical contexts.

But again ,I think we are unable to calculate something about waves ,for instance , the weight of light (wave) and/or the mass of voice waves may not be calculated, however this will be very important according to this thread's goal ,whether the exact equation of voice wawe's are writable

Edited September 4, 2016 by blue89

[studiot]

The (linear) wave equation in one dimension is given by a differential equation

 

[math]\frac{{{\partial ^2}y}}{{\partial {t^2}}} = {v^2}\frac{{{\partial ^2}y}}{{\partial {x^2}}}[/math]

With (one) commonly used solution

[math]y = a\sin \frac{{2\pi }}{\lambda }\left( {x - vt} \right)[/math]

As

As you can see it connects space and time.

The standing wave obeys the differential equation but has a solution

[math]y = 2a\sin \left( {\frac{{2\pi x}}{\lambda }} \right)\cos 2\pi nt[/math]

This separates space and time so for any time t you can plot the value of y over x.

Edit

Doing this with the wave equation is the beginnings of quantum theory by the way.

Edited September 4, 2016 by studiot

[blue89]

hi ;

 

Really I think time parameter might have various qualifications. when I read a piece of a paper of stephen hawking , I realised that he uses some words that makes us almost sure/confident that still we have no obvious useful information about both time characater and >3D dimension.

 

for instance he uses a sentence like this

"We predict that >3D would be...."

the red coloured words proves this us.

the thing that I am sure is the difficulty of finding such implied properties or new things about this issue.

but at the same time ,I think this is clear that we are able to find something very interesting and useful via using different methods.

Note please : I am sometimes using cryptologies (and all of my studies are new types (not traditional))

 

 

 

 

Edit

Doing this with the wave equation is the beginnings of quantum theory by the way.

this reminiscents me somethings are GREAT!

 

Edited September 5, 2016 by blue89