Post by: guest39538 on 29/05/2016 08:31:34
Post by: evan_au on 29/05/2016 12:00:01
You can also add and subtract vectors.
You can also do rotations (around n different axes), and translation (moving along n different axes).
Under some conditions you can also multiply and divide vectors, but you need to be very careful that you are using consistent dimensions (eg length, velocity, acceleration, angular momentum); usually multiplying and dividing changes the units, while addition, subtraction and scalar multiplication does not usually change the units.
And if you have a set of n equations in n dimensions, you can also (sometimes) identify an Inverse.
What would you like to prove or analyse using n-dimensional vectors?
Post by: guest39538 on 29/05/2016 15:02:52
What would you like to prove or analyse using n-dimensional vectors?
That the size,shape and age of the Universe is n.
t=
=nd=
=nx=y=z=n
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By "expansion and contraction", I assume you are talking about multiplying a vector by a scalar?
I mean things that expand +ve away from a singularity point of 0 and things that contract -ve to a singularity point of 0. Light contracts and expands for example.
Or did you mean me doing this
+E=>4/3 pi r³
-E=<4/3 pi r³
Post by: jeffreyH on 29/05/2016 22:10:27
http://www.slimy.com/~steuard/teaching/tutorials/Lagrange.html (http://www.slimy.com/~steuard/teaching/tutorials/Lagrange.html)
Post by: jeffreyH on 29/05/2016 22:34:48
+E=>4/3 pi r³
-E=<4/3 pi r³
I assume that what you mean is that for any given volume of space the positive energy is continually expanding out of the surface while negative energy is compressing into this same volume through its surface. This may well be an apt description of the effects of dark energy.
Post by: puppypower on 29/05/2016 23:12:35
Only expansion and contraction exists in n-dimensional vector analysis?
There is pure science and there is applied science. Applied science extrapolates pure science to anticipate and extrapolate things that may not be natural. The CERN particle collider is not natural, but is abased on many areas of applied science. All the science needed for the entire collider assembly has a foundation in pure science. However, applied science is the framing, roof and siding; meat on these bones.
The concept of direction is a very useful applied science concept. Nature does not care about up or down, left or right or north and south. These are useful human conventions, used by applied science, to extrapolate pure science; navigation and bridge building. An electron and proton will interact in all directions even if you flip and rotate it. Gravity is only concerned with mass and distance and not direction. Direction is more useful to humans since the vectors of direction tells us the forces needed for propulsion and the strength of materials. Knowing the direction, north, from the Northstar, is useful for navigation; applied, but the wind and currents have no needs this.
One interesting application of applied science is magic. A magic trick, such as levitation, requires a wide range of applied science, from structural science to psychology. The basis of the needed materials and the human brain will be grounded on pure science. These pure principles are extrapolated, beyond natural, to create the magic trick. To those who don't know there is a difference between pure and applied, they can be fooled into believing applied science; composite trick, is the same as pure science. In this case, levitation becomes a law of nature.
Let me give a classic example of what can happens if there is confusion between pure and applied science. The metal aluminum, will never be found free in nature; earth. Aluminum metal is way too reactive with oxygen. To make aluminum metal, we would need to use applied science; electricity, to make it in the lab or factory.
Say in the very early days of development, when this process was a trade secret, some of the aluminum metal, made in the lab, was taken home and lost in the woods. A geologist finds the metallic aluminum, leading to excitement and buzz in the geology world. This should not be possible, based on existing theory, yet it was found in the woods. It sort of a magic trick; applied being called pure. However, is not a trick, per se, because this was never intended. It was a mistake that has to be covered up at risk of losing your job and being sued.
Because this is such a provocative discovery, the geologists call in the geo-physical chemists, who speculate and then they simulate how the earth could have made metallic aluminum. One of the key researcher is successful. His discovery is also applied science. However, since it simulates a reasonable scenario, for this discovery, this is also assumed to be pure science and a natural path of nature. This acceptance, now has other implications. It can explain others things, as well as bring different things into question. Everyone likes the new girl on the block; latest science fad. So more studies are done with even more applied science, called pure, creating an image of the earth in the image of man.
The moral of the story is it is useful to separate pure and applied science. Direction has far more practical use for man.
Post by: evan_au on 29/05/2016 23:43:41
Quote from: TheBox
the size,shape and age of the Universe is n.
t=...=n
d=...=n
x=y=z=n
x, y, z and t are dimensions. In conventional notation, x, y & z measure distance, while t measures time.
n is a number (a scalar). It counts dimensions. So the point (x, y, z, t) has a location in n=4 dimensional space (it is a vector).
x, y, z & t are (fairly) independent - you can move along the x axis without moving along y or z.
The dimension of time (t) is a bit different from the dimensions of space, in that the speed of light is a limitation on your movement. And while we can do a rotation about the z-axis, we don't know how to do a rotation about the time axis.
You can look at points having x=1 mm, or 3.14159 km or -10 parsecs or +10 billion light years.
But you can't say "x=n" or "t=n", because the units of x are distance and the units of t are time; the units of n are a count of the number of dimensions. The units aren't compatible. You can't say 3 apples=4 orangutans.
Post by: guest39538 on 30/05/2016 06:25:27
Quote from: TheBoxthe size,shape and age of the Universe is n.
t=...=n
d=...=n
x=y=z=n
x, y, z and t are dimensions. In conventional notation, x, y & z measure distance, while t measures time.
n is a number (a scalar). It counts dimensions. So the point (x, y, z, t) has a location in n=4 dimensional space (it is a vector).
x, y, z & t are (fairly) independent - you can move along the x axis without moving along y or z.
The dimension of time (t) is a bit different from the dimensions of space, in that the speed of light is a limitation on your movement. And while we can do a rotation about the z-axis, we don't know how to do a rotation about the time axis.
You can look at points having x=1 mm, or 3.14159 km or -10 parsecs or +10 billion light years.
But you can't say "x=n" or "t=n", because the units of x are distance and the units of t are time; the units of n are a count of the number of dimensions. The units aren't compatible. You can't say 3 apples=4 orangutans.
Now you have confused me, n is any natural number , I am not using apples and pairs,
because the units of x are distance and the units of t are time and the value is any n?
d=
this says the distance from (a) to ''nowhere'' is (n) any number, zero point source, the ''blackness'' of space in any direction or in the ''dark''.
Post by: guest39538 on 30/05/2016 06:26:11
+E=>4/3 pi r³
-E=<4/3 pi r³
I assume that what you mean is that for any given volume of space the positive energy is continually expanding out of the surface while negative energy is compressing into this same volume through its surface. This may well be an apt description of the effects of dark energy.
any volume of space or any volume of ''matter'', and from the centre not just the surface.
Post by: evan_au on 30/05/2016 10:58:49
Quote from: TheBox
n-dimensional vector analysisAccording to the original question, n is a count of the number of dimensions, eg 1 dimension for a ruler, 2 dimensions for a sheet of paper (or the surface of the Earth), 3 dimensions for space (or the volume of the Earth), 4 dimensions for Einsteins's spacetime, etc.
n will be a whole number (unless you are describing some fractal system (https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension)).
The units of n are the number of dimensions for whatever space you are discussing.
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this says the distance from (a) to ''nowhere'' is (n) any number, zero point source
This notation seems to be representing "d" as a distance vector from point "a" to another point. But this point cannot be nowhere, or the vector will be meaningless. It must be to somewhere.
But it is a bad choice of notation to call this second point "n", because you started by defining n as the number of dimensions.
So call the end point of the distance vector something else, like "b": "d" is a distance vector from point "a" to another point, "b".
The position of "a" and "b" is generally defined relative to some arbitrary zero point, eg Greenwich, the equator, the Sun, or the center of town.
The length of "d" will in general be a real number (unless you construct "d" to be an exact integer in length). The units of d will be some units of length, eg Angstroms, furlongs, Astronomical Units, cubits, miles or meters (personally, I prefer meters).
Quote
to ''nowhere'' is ... the ''blackness'' of space in any direction or in the ''dark''.You've lost me here.
If point "b" is in the middle of the Sun, and you started at point "a" and then moved distance "d", you would not be in the blackness of space, but in an uncomfortably bright location.
The distance d does not determine the brightness. If you start off at a different point (call it "c") then move distance "d", you will end up at a different point which may happen to be dark - or it may be bright.
As I understand it, the intensity of illumination is an example of a scalar field (https://en.wikipedia.org/wiki/Scalar_field), which varies as you move through the n dimensions. Some points (like in the middle of the Sun) have a very high intensity of light, while others (like half way to Alpha Centauri) have a much lower level of illumination. You would not call intensity of illumination a vector.
What would you like to prove or analyse with you new multi-dimensional tools?
Post by: jeffreyH on 30/05/2016 11:51:55
Post by: guest39538 on 30/05/2016 16:04:47
Quote from: TheBoxn-dimensional vector analysisAccording to the original question, n is a count of the number of dimensions, eg 1 dimension for a ruler, 2 dimensions for a sheet of paper (or the surface of the Earth), 3 dimensions for space (or the volume of the Earth), 4 dimensions for Einsteins's spacetime, etc.
n will be a whole number (unless you are describing some fractal system (https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension)).
The units of n are the number of dimensions for whatever space you are discussing.
I think n-dimensional is another ambiguous meaning, there is many explanations on internet search.
So what would represent any number from 0 to ''infinite'' if not n-dimensional?
I firstly want something that represents any ''length''.
Quote from: box
this says the distance from (a) to ''nowhere'' is (n) any number, zero point source
Quote
This notation seems to be representing "d" as a distance vector from point "a" to another point. But this point cannot be nowhere, or the vector will be meaningless.That is correct x=
But not to another point source, to the ''blackness'' of space (next to a ''light'' reflecting body) , the space you see ''nothing''. My vector is not a point to point vector of A to B, B is not there , it is any ''zero'' point of space.
Look up into the night sky, you see stars, next to the stars you see relative ''blackness'' , my vectors travel into this ''blackness'' , hence why I am putting
n being any distance and any ''age'' . Quote
The length of "d" will in general be a real number
Only if the length has two point sources, see above and please try to understand.
Post by: Bored chemist on 30/05/2016 18:02:42
Post by: guest39538 on 30/05/2016 19:23:38
Is anyone sure what Thebox is talking about, or is it horribly unclear?
It is as clear as space, which part do you not understand?
Post by: evan_au on 30/05/2016 22:27:55
Quote from: JeffreyH
You could have a vector <x y z L> that records the intensity of illumination at various points in space. In this way luminosity could be a component of a vector.What makes it an n-dimensional vector is if the n dimensions are independent.
So you need to ask a question like: "In 3-Dimensional space, can I vary Luminosity L without varying Position x, y or z?".
I would suggest that:
- if you change x so that you move from the center of the Sun to the Oort cloud, Luminosity L will change dramatically.
- If point S=(x, y, z) identifies a point inside the Sun, then Luminosity L will be fixed, and won't change unless you change x, y or z (or time - the Luminosity inside the Sun does change with time).
By the way, the units of light intensity would be something like Watts/m2, Candelas or Lumens (I get these units confused). Not a scalar!
Quote
So scalars can participate as the components of a vector.I agree that this is possible in pure mathematics, but it is less common in Physics.
Most quantities in Physics have some dimensions. So for example:
- if you wish to describe the position of the Earth relative to the Sun in Cartesian coordinates, you could conveniently use units of "Millions of km" (although other distance units are also acceptable, and can be converted into "Millions of km").
- if you wish to describe the velocity of the Earth around the Sun in Cartesian coordinates, you could conveniently use units of "km/sec" (although other units of speed are also acceptable, and can be converted into "km/sec").
- If you describe the position of the Earth in Polar coordinates, there will be a distance in "Millions of km", plus an angle in degrees or radians
- If you describe the orbit of the Earth as an ellipse with a certain eccentricity, the eccentricity is dimensionless, but it is turned into a real figure by the semi-major axis, which has units of "Millions of km" (for example). By the way, describing the orbit of the Earth takes more than 3 dimensions, since it involves an element of time (in fact, you need 6 dimensions (https://en.wikipedia.org/wiki/Orbital_elements) to uniquely define an orbit).
So I agree that scalars can be components of a vector, but in Physics it is the exception, rather than the rule.
Post by: evan_au on 30/05/2016 22:59:39
Quote from: TheBox
I think n-dimensional is another ambiguous meaning, there is many explanations on internet search.So pick one of the well-established definitions instead of trying to make up your own.
Once you have absorbed this definition, it shows you how to add and subtract vectors, and then you can then start to learn more:
https://en.wikipedia.org/wiki/Euclidean_vector
It is an extremely flexible concept, with standard techniques. Use the standard techniques instead of inventing new ones.
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So what would represent any number from 0 to ''infinite'' if not n-dimensional?A scalar can represent any number from 0 to infinity (or even negative infinity). A scalar is sometimes described as zero-dimensional.
A 1-diminsional vector can also represent any number from 0 to infinity (or even negative infinity).
You don't need to go n-dimensional for this. You can just use High-School arithmetic.
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I firstly want something that represents any ''length''.A 1-dimensional vector (with units of meters) can represent any length from 0 to infinity (or even negative infinity).
If you want to uniquely describe a point in 3-Dimensional space, you need 3 such numbers (x, y and z). This is a valid application of n-dimensional, with n=3.
Quote from: TheBox
it is any ''zero'' point of space.
A zero reference point in space is often selected arbitrarily, usually for a practical reason like making the calculations easier.
But once you have selected your zero point, it is best not to change it during your calculations, or you will be even more confused.
The zero point (or "Origin") is labelled something like Origin O=(0, 0, 0) in 3-dimensional space.
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this says the distance from (a)
You assume the existence of a point A, not at the Origin O. It's location requires the specification of 3 dimensions, for example A=(xA, yA, zA).
You can treat A as a 3-dimensional vector, from the Origin O to Point A.
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My vector is not a point to point vector of A to B, B is not there.
You talk about a Distance vector D. You calculate the Distance vector by doing a subtraction in n-dimensional space.
Lets assume that we don't know where B is (since it's obviously not where I thought it was). So we can label it as B=(xB, yB, zB), where xB, yB and zB are all unknown.
You calculate vector D = B - A = (xB-xA, yB-yA, zB-zA).
And this works no matter where B is.
Post by: Alan McDougall on 30/05/2016 23:34:40