What is a field?

[studiot]

I was not going to, but, Xerxes, since you bring it up

 

A field in Physics can entail two quite distinct and different coordinate systems and usually does.

No transformation exists between these coordinate systems.

 

I do think such discussion is way beyond the basics suitable for this thread, but I will elaborate if you wish.

[Mordred]

 

I do think such discussion is way beyond the basics suitable for this thread, but I will elaborate if you wish.

Though it is refreshing

[Nedcim]

 

Behaviour. Define the structure or form of a photon or electron.

 

Behavior is dependent on the structure. Why define anything by dependent factors?

I should have said physics describes behaviour.

 

 

True. Wouldn't you also say that physics deals with the structure of matter?

 

 

One thing that comes out of Green's / Gauss' theorems as a surprise to many is the fact that ie form and structure for many physical phenomena, in the interior of a bounded region, can be completely defined by knowledge of what is happening on the surface.

 

Citation? Anyhow the surface is still defined by the structure.

[geordief]

 

 

Think of it this way.

 

Consider a point mass, m at point A in a potential (energy) field.

 

Moving it to a point B of lower energy yields some energy for some purpose eg accelerating the mass.

 

If we now complete the circle we have to re-supply that energy to the mass to return it to point A because the PE field is a conservative field.

 

But at all time the potential is there at both points A and B and all points in between.

Is this (the part I have bolded) what happens when an ice dancer folds her arms and spins faster as a result?

[studiot]

 

 

Citation? Anyhow the surface is still defined by the structure.

 

Since you ask, try

 

The Boundary Element Method

 

Brebbia.

 

 

Yes, the one defines the other.

 

The point is that it can be easier to perform the calculations over a few boundary elements than many internal mesh points.

Integration over a closed surface or loop is often computationally easier than a volume integral.

 

Do you need references for this stuff?

Is this (the part I have bolded) what happens when an ice dancer folds her arms and spins faster as a result?

 

The angular momentum remains constant but the components change.

 

That is as the radius decreases the rotational speed increases.

 

This is not an energy phenomenon.

 

Moving a particle from point A to B removes potential energy from the particle or changes it to kinetic energy.

 

A good example would be the drop hammer on a piledriver.

 

The gravitational energy lost by the hammer in dropping is transferred to the pile.

This is replaced when the cycle is completed by raising the hammer back to drop height.

Edited February 3, 2017 by studiot

[geordief]

 

Moving a particle from point A to B removes potential energy from the particle or changes it to kinetic energy.

 

A good example would be the drop hammer on a piledriver.

 

The gravitational energy lost by the hammer in dropping is transferred to the pile.

This is replaced when the cycle is completed by raising the hammer back to drop height.

Is there a kinetic energy field which is a "mirror image " of the potential energy field ?

Edited February 3, 2017 by geordief

[studiot]

What exactly is a kinetic energy field?

 

You have a potential energy field because a massive ( massive = possessing mass) body exerts a gravitational effect on every point in the region around it.

 

There is something called 'complementary energy' in continuum mechanics but it is not really a mirror image.

[geordief]

What exactly is a kinetic energy field?

 

You have a potential energy field because a massive ( massive = possessing mass) body exerts a gravitational effect on every point in the region around it.

 

There is something called 'complementary energy' in continuum mechanics but it is not really a mirror image.

A map of the kinetic energy of a test body in the region of (in this case ) a massive body ?

 

That was my idea for what it was worth.

 

I did google the term to see if there was such a thing as a "kinetic energy field" (and there seems to be) but I know nothing about them apart from that they seem to exist as a concept.

Edited February 3, 2017 by geordief

[Mordred]

Were hitting some difficult concepts to properly explain without overwhelming those who aren't familiar with field theories. So lets step back from the more complex mathematics posted thus far and review some basics.

 

First off as a time saver please read this article on work and energy.

http://galileoandeinstein.physics.virginia.edu/142E/10_1425_web_ppt_pdfs/10_1425_web_Lec_12_WorkandEnergy.pdf its important to understand how work is defined in terms of the vector paths. There is numerous key aspects to understand on path dependent work. Even more complex the explain is divergence theorem which includes Stokes theorem,Gauss theorem and Greens theorem. These are in essence calculus subjects covered with vector analysis.

 

https://betterexplained.com/articles/divergence/

 

this link is extremely rudimentary. I obviously cannot teach entire chapters on a forum lol, so the best we can do is supply informative links

 

This link will show just how much is truly involved in the above

http://www.uvm.edu/~cems/keoughst/LectureNotes141/Topic_02_(VectorAnalysis).pdf

 

Needless to say it takes a considerable amount of time to study Studiot may know of far better online resources to learn this stuff. I obviously can't post my textbooks lol.

 

Now the above also relates to extremely important principle. The principle of least action. Which equates kinematic motion involving potential energy and kinetic energy. Feyman lectures has a decent rudimentary coverage of this

 

http://www.feynmanlectures.caltech.edu/II_19.html

 

Now if a force over a given simply connected region of space S (continuous ie no holes, ie not a donut). Can be expressed as the negative gradient of a scalar function.

[latex]F=-\nabla\varphi[/latex] the minus sign denotes a conservative force

 

now we want to know when a scalar potential function exists to answer this we need two other relations as equivalent to the above equation.

[latex]\nabla*F=0[/latex]

and

[latex]\oint F\cdot dr=0[/latex]

 

for every closed path in our simply connected region S. We proceed to show that each of these three equations implies the other two. Lets start with

[latex]\nabla*F=0[/latex]

then [latex]\nabla*F=-\nabla*\nabla\varphi=0[/latex]

 

Now were going to need some further details that is going to take a bit lol The Laplacian of a potential.

 

[latex]\nabla*\nabla\varphi=\begin{pmatrix}\hat{x}&\hat{y}&\hat{z}\\ \frac{\partial}{\partial{x}}&\frac{\partial}{\partial{y}}&\frac{\partial}{\partial{z}}\\\frac{\partial\varphi}{\partial{x}}&\frac{\partial\varphi}{\partial{y}}&\frac{\partial\varphi}{\partial{z}}\end{pmatrix}[/latex]

 

by expanding the determinant we obtain

[latex]\nabla*\nabla\varphi=\hat{x}(\frac{\partial^2\varphi}{\partial y \partial z}-\frac{\partial^2\varphi}{\partial z \partial y})+\hat{y}(\frac{\partial^2\varphi}{\partial z \partial x}-\frac{\partial^2\varphi}{\partial x \partial z})+\hat{z}(\frac{\partial^2\varphi}{\partial x \partial y}-\frac{\partial^2\varphi}{\partial y \partial x})[/latex]

 

With the first two equations and the last equation we get

 

[latex]\oint F\cdot dr=-\oint\nabla\varphi\cdot dr=-\oint d\varphi[/latex]

 

Now [latex]d\varphi[/latex] gives way to [latex]\varphi[/latex] since we have specified a closed loop and we get zero for every closed loop path in our region S. AS long as we are dealing with an uncharged scalar field does this above analysis apply.

 

So the only two points we need to concern ourselves with is the start and finish end points regardless of path taken ,start point A, end point B

 

So work done by force is

[latex]\int^b_a F\cdot dr=\varphi (a)-\varphi (b)[/latex]

 

Neat we immensely simplified all the above

 

now lets quickly show Newtons law

[latex]F_g=-\frac{Gm_1m_2\hat{r}}{r^2}=\frac{k\hat{r}}{r^2}[/latex]

 

as the force is radially inward integrating the first equation from infinity to r we obtain

 

[latex]\varphi_g r-\varphi{\infty}=-\int^r_\infty F_g\cdot dr=+\int^\infty_r F_g\cdot dr[/latex]

 

from this last equation we can see that the vector components is shown under a change in sign

 

so when you have gravity the integral on the RHS of the last equation is negative so

 

[latex]\varphi_g r=-\int^\infty_r\frac{k dr}{r^2}=-\frac{k}{r}=-\frac{Gm_1m_2}{r}[/latex] the negative sign denotes gravity is attractive. k is a curvature constant ie negative curvature (gravity well). I'll let you think about that and the term manifold in regards to the above.

 

The above is a lesson from Arfkens " Mathematics for physicists" It is an excellent textbook that doesn't deal directly with any model such as GR, QM etc but teaches the required math to understand those models

 

https://www.amazon.ca/Mathematical-Methods-Physicists-Comprehensive-Guide/dp/0123846544

Edited February 3, 2017 by Mordred

[studiot]

Thank you Mordred for that large list of net references.

 

Since I normally go with books, here are a couple to complement them.

 

Firstly

 

The Standard classic for the heavier mathematics you refer to from the founder of this sort of theory, O,D, Kellogg

 

Foundations of Potential Theory

 

O D Kellogg

 

Kellogg not only develops the mathematical theory necessary for those fields that are amenable to Potential Theory (ie have Scalar, Vector or Tensor Field Potentials) but also offers much insight into the connections with some of the more esoteric pure maths behind it such as The Heine Borel Theorem.

It is strictly an analytical approach in the mathematical analysis sense.

 

Secondly this delightful tome from the Engineering sector for those using the techniques rather than developing them

 

Engineering Field Theory

 

A J Baden Fuller

 

Probably more appropriate here, this book offers not only practical insights into using the theory and solving the equations both analytically and numerically since engineers need to use equations rather than study their pedigree.

It covers a wide range if Fields used in practice and includes a large table of the relevant formulae from all the different chapters about different sorts of Fields.

I offer the following brief extracts that may be of help here.

 

First the chapter list

 

 

Then two pages from the introduction that all here can read and hopefully draw something from

 

[Bender]

A map of the kinetic energy of a test body in the region of (in this case ) a massive body ?

 

That was my idea for what it was worth.

 

I did google the term to see if there was such a thing as a "kinetic energy field" (and there seems to be) but I know nothing about them apart from that they seem to exist as a concept.

It is perfectly possible to define some kinetic energy fields. What you may have found on google are kinetic energy fields in a fluid, where the kinetic energy density in each location in the fluid is given. You could also define a kinetic energy density field in a rigid body, but I don't see how that would be useful.

 

 

For those wanting to know more about Green's theorem and boundaries: in words it says "what comes in minus what goes out is what changes inside".

A simple example: consider the temperature gradients on the surface of an object. Now add together all the heat that flows in through the surface of the object, minus all the heat that flows out. Now you have the change in thermal energy of the object, from which you can calculate the mean temperature change. In short: all you need to know to calculate internal temperature change is how much heat passes the surface.

[studiot]

It is perfectly possible to define some kinetic energy fields. What you may have found on google are kinetic energy fields in a fluid, where the kinetic energy density in each location in the fluid is given. You could also define a kinetic energy density field in a rigid body, but I don't see how that would be useful.

 

 

For those wanting to know more about Green's theorem and boundaries: in words it says "what comes in minus what goes out is what changes inside".

A simple example: consider the temperature gradients on the surface of an object. Now add together all the heat that flows in through the surface of the object, minus all the heat that flows out. Now you have the change in thermal energy of the object, from which you can calculate the mean temperature change. In short: all you need to know to calculate internal temperature change is how much heat passes the surface.

 

 

Sometimes its good to bring some simple sanity to a thread. +1

[Xerxes]

A field in Physics can entail two quite distinct and different coordinate systems and usually does.

No transformation exists between these coordinate systems.

Please explain.

[mistermack]

Studiot, thanks for posting the Baden-Fuller exerpt.

It told me what I wanted to know in a few sentences.

 

If the rest of the book is as clear, it really is a delightful tome.

Edited February 3, 2017 by mistermack

[studiot]

Please explain.

 

Sure,

 

I originally said that a field 'exists' in a region of space, that you later called a manifold.

So we need a coordinate system to identify each point in the region or manifold.

 

The field function then assigns a particular value to each point we identify for the field variable.

 

In Physics that field variable may need only a numerical scale or it may need a whole coordinate system of its own.

 

So a temperature field assigns a number temperature on a temperature scale to each point and this scalar field thus has a one dimensional field variable.

 

A more complicated field variable, say a fluid flow vector, will require another separate coordinate system to be established at every point in the region, in order to be able to define the vector.

 

There may be a transformation to allow us to move between the coordinate systems.

This will happen if both are of suitable physical quantities.

If this is the case the transformation is called a chart.

There is a chart for each point in the original coordinate system and the entire portfolio of charts is called an atlas.

There will also then be a relationship between each pair of charts in atlas. This transformation is called a Connection.

 

Of course it may also be that there is no transformation so it is impossible to turn temperatures into spatial positions, and vice versa.

 

 

Studiot, thanks for posting the Baden-Fuller exerpt.

It told me what I wanted to know in a few sentences.

 

If the rest of the book is as clear, it really is a delightful tome.

 

Glad you are finally happy.

 

I secretly hoped you would read it.

[Mordred]

Good and well appropriate excerpts you included Studiot. Definetely well appropriate complement to the links I posted. I figured you would have a useful reference handy. Like you I prefer the textbooks to articles.

 

I'm also giving +1 to Benders last post good example. Side link comment on that example but that example also applies in a sense to the holographic principle

Edited February 3, 2017 by Mordred

[Xerxes]

So we need a coordinate system to identify each point in the region or manifold.

Yes, although in the abstract sense the "point" IS just the local coordinates

 

The field function then assigns a particular value to each point we identify for the field variable.

This is not the best definition of a field - it relies heavily on the infamous Axiom of Choice, but yes, it will do

 

In Physics that field variable may need only a numerical scale or it may need a whole coordinate system of its own.

Here we disagree. Coordinates are a property of manifolds. If you are thinking of say a vector field, then sure, each field element must be described relative to something. These are called "basis" vector" not coordinates. But, for a vector tangent to a manifold (this makes sense even for boring manifolds like [math]R^n[/math]) these basis vectors are derived from the coordinates of the manifold in question.

For differentiable manifolds these are called directional derivatives (or differential operators) and take the form [math]\frac{\partial}{\partial x^k}[/math] where the [math]x^k[/math] are the coordinates at the point the vector is applied

 

So a temperature field assigns a number temperature on a temperature scale to each point and this scalar field thus has a one dimensional field variable.

Yes, but the point at which this scalar is applied has the same coordinates as ever.

 

A more complicated field variable, say a fluid flow vector, will require another separate coordinate system to be established at every point in the region, in order to be able to define the vector.

Here I disagree - see above

 

There may be a transformation to allow us to move between the coordinate systems.

This will happen if both are of suitable physical quantities.

I covered this in an earlier post. Coordinate transformations refer to points.

If this is the case the transformation is called a chart.

No it isn't. A chart consists of a subset of the manifold together with its continuous and invertible mapping to an open subset of[math]R^n[/math]

[mistermack]

 

Glad you are finally happy.

 

I secretly hoped you would read it.

Happy-errr. One question leads to another, so I'm never happy.

 

I said on page 2 that I was beginning to suspect that my answer was "nobody knows". And the Baden-Fuller introduction pretty well confirmed that. You don't postulate three different hypothetical scenarios, if you know what something is.

The term "field" itself doesn't help, as people seem to have varying aspects as to what it means.

 

You wrote "The field is just a catalog of the values of the field variable in some region of space at a particular point in time."

 

I was thinking of a field as the volume of space combined with the magnetic or gravitational effects within it.

Wikipedia in it's "Magnetic Field" page has as the first sentence "A magnetic field is the magnetic effect of electric currents and magnetic materials. " In other words, not a

the catalog, but the actual thing that is catalogued.

 

It was in that sense of field that I asked the question.

But now one question has turned into about half a dozen. That's the way it goes I guess.

[studiot]

Yes, although in the abstract sense the "point" IS just the local coordinates

 

No because the coordinates depend upon the coordinate system. The point remains the same throughout, just as the field variable at that point remains the same, with the same values regardless of the coordinate system used to place it.

 

This is not the best definition of a field - it relies heavily on the infamous Axiom of Choice, but yes, it will do

 

Right at the outset, I noted that the definition of a Field in Physics is not the same as the definition in Mathematics and that we should use the one in Physics since this question was posed in the Physics section classical at that so all QM considerations are theoretically off topic.

 

Here we disagree. Coordinates are a property of manifolds. If you are thinking of say a vector field, then sure, each field element must be described relative to something. These are called "basis" vector" not coordinates. But, for a vector tangent to a manifold (this makes sense even for boring manifolds like [math]R^n[/math]) these basis vectors are derived from the coordinates of the manifold in question.

For differentiable manifolds these are called directional derivatives (or differential operators) and take the form [math]\frac{\partial}{\partial x^k}[/math] where the [math]x^k[/math] are the coordinates at the point the vector is applied

 

Yes we do disagree.

The directional derivatives are just that. Basis vectors for the direction part of of the vector at the location concerned.

They provide no information as to the magnitude of the field variable.

I could place a fluid velocity of any (reasonable) magnitude I chose at a point.

 

Yes, but the point at which this scalar is applied has the same coordinates as ever.

 

So what? the value of the scalar is not determined by position.

Have you ever plotted a scalar temperature field?

Here I disagree - see above

 

So you define a physical magnet as a field?

 

I covered this in an earlier post. Coordinate transformations refer to points.

 

Coordinate transformations refer to the conversion between the coordinates of points when measured in one coordinate system or another. It says nothing about the points themselves.

In some cases inverse transformations are not possible.

 

No it isn't. A chart consists of a subset of the manifold together with its continuous and invertible mapping to an open subset of[math]R^n[/math]

 

We will agree to differ here, but this is really off topic math.

But I would observe that quote open subset of[math]R^n[/math] /quote defines a second coordinate system, exactly as I said.

 

The last time I tried to quote and delete unwanted sections of a post to make the replies relevant it broke the forum programming.

 

So apologies for writing my replies into your post.

Edited February 3, 2017 by studiot

[Mordred]

Thats not surprising the subject of fields is immense. There is a 1000 plus article I've spent nearly two years studying and I'm still only part way through it.

 

http://arxiv.org/abs/hepth/9912205 : "Fields" - A free lengthy technical training manual on classical and quantum fields

[studiot]

 

I was thinking of a field as the volume of space combined with the magnetic or gravitational effects within it.

Wikipedia in it's "Magnetic Field" page has as the first sentence "A magnetic field is the magnetic effect of electric currents and magnetic materials. " In other words, not a

the catalog, but the actual thing that is catalogued.

 

Yeah that's about it alright except that the thing that is cataloged may have physical embodiment or may be just a catalog of numbers, like a velocity field.

 

But yes we are very loose in mixing up the measurement of something with the thing itself in lots of circumstances.

 

Edit

 

Here are some thoughts on why we should not include the region of space in the field.

 

1) Consder the field of an electromagnet. If we turn the magnet off the field disappears, but the region does not.

 

2) What if we have more than one field in the same region of space eg a direction field and velocity field in a fluid?

 

3) What if we change the field values but not the region eg the temperature field in a bar changes if I heat one end.

 

4) What if I change the boundaries of a region of space? The values may alter or may remain the same depending upon the type of field

eg both a fluid and electrostatic field will expand to fill expanded boundaries. The fluid values will, in general change but the electrostatic ones will not.

Edited February 3, 2017 by studiot

[Xerxes]

The directional derivatives are just that. Basis vectors for the direction part of of the vector at the location concerned.

 

They provide no information as to the magnitude of the field variable.

But a (tangent) vector is written as, say, [math]v_p=\sum\nolimits_k \alpha^k \frac{\partial}{\partial x^k}[/math] where the [math]\alpha^k[/math] are scalars. These give "magnitude" to the "direction" along each of the coordinates.

This is a standard definition - open a good text.

 

The idiosyncratic format of your response to me makes it impossible for me to say more

[studiot]

But a (tangent) vector is written as, say, [math]v_p=\sum\nolimits_k \alpha^k \frac{\partial}{\partial x^k}[/math] where the [math]\alpha^k[/math] are scalars. These give "magnitude" to the "direction" along each of the coordinates.

This is a standard definition - open a good text.

 

The idiosyncratic format of your response to me makes it impossible for me to say more

 

 

You are just dodging the issue.

 

Where is the scale axis the the scalar constants a^k lie on, for it is not parallel to any of the existing axes x^k.

 

Try this practical demonstration.

 

Take a piece of gridded paper, draw a pair of ordinary axes and mark the point 5,5.

 

Now at the point 5,5 draw a vector of magnitude [math]\sqrt 2 [/math] at 45^oso that it takes up zero room on the paper,

 

because one thing is for certain

 

That vector does not extend from x =5 to x = 6 and y = 5 to y = 6

 

You require a new coordinate system in a new space, joining the old coordinate system at one point only, to be able to accomplish this plot.

 

 

[geordief]

 

 

You are just dodging the issue.

 

Where is the scale axis the the scalar constants a^k lie on, for it is not parallel to any of the existing axes x^k.

 

Try this practical demonstration.

 

Take a piece of gridded paper, draw a pair of ordinary axes and mark the point 5,5.

 

Now at the point 5,5 draw a vector of magnitude [math]\sqrt 2 [/math] at 45^oso that it takes up zero room on the paper,

 

because one thing is for certain

 

That vector does not extend from x =5 to x = 6 and y = 5 to y = 6

 

You require a new coordinate system in a new space, joining the old coordinate system at one point only, to be able to accomplish this plot.

 

 

Should I be paying special attention here as regards to your ( earlier but recent ) in a separate question?

 

http://www.scienceforums.net/topic/101620-intrinsic-curvature/page-1#entry961025

 

Are the two threads connected at this point? (irony intended )

Edited February 4, 2017 by geordief

[mistermack]

Earlier in the thread, I asked, if the magnetic force is provided by photons, why we don't see them, or detect them streaming out of a magnet.

Nobody answered that one, but I'm guessing that the answer is that they are virtual photons, discussed here:

 

So it's another "we don't know, but this works" answer. They are mounting up.