Mordred

If you plot the degree changes on each plane then the rate of degree change is identical, the only difference is the two simultaneous direction changes for each plane.

I find I'm a little confused here. The article is excellent for describing the principles of axis for you. However you are still looking for one single function that would control those three principles of axis. The problem is the only value shown in the video is e. Which is a common connection point on the animation. Part of the problem here is the animation only shows one number for e. E being the small connection circle on the animation which follows a tilted circular axis. However Hans doesn't show the coordinate changes of e, and only shows one the degree changes made as e moves. So I assume he's showing the degree change of e as it follows its tilted circular path To add complication all three rotating planes have three different relation changes to (let's call it reference e). One plane undergoes an additional tilting compared to the other two planes, and e doesn't maintain the same location on another plane but cuts through that plane in a circular arc. Yet your asking everyone to build one function that describes this? Especially when there is far more going on in the program than what is shown in the animation. Looking at this the clue is in the plane that e cuts through the plane. This is the plane that involves both degree changes simultaneously ( though all three planes do ) the point where e resides on that particular plane coincides with e along the tilted circular path. Degrees between the y to x axis and degrees to z to x axis. The change to each is equal and simultaneous.

This last post is based on a misconception that expansion or contraction requires a center. Which is simply wrong.

It's never a good idea to attempt to answer questions on a forum based on your personal feelings or arguments. For one thing expansion has very little to do with dark matter. Even if the universe was contracting you would still need dark matter for galaxy rotation curves and early large scale structure formation. Secondly scientists didn't say ooh lets have dark energy and make the universe expand. Measurement data shows the universe is expanding in both distance and temperature thermodynamic relations. However the universe doesn't require dark energy to explain how a universe expands. Its needed to explain why that expansion isn't slowing down.

That's a nice article, thanks Studiot.

Most trigonometric applications are functions but they are not necessarily written as f(x). For example an exponent function is F(x)=b^x. So when see an exponent your looking at a type of function. I think it's safe to assume much of your discussion on the math forum has been discussion on what constitutes a function. Particularly rereading Studiots post.

Well this last document clears up a lot. What you have in the document is viable. I recall seeing something similar though I'll have to dig for it. Though it's essentially a trigonometric application. I don't see why you want direction when your returning the magnitude of the vector. Though you can also calc each vector with the necessary details. Ps this isn't anywhere close to what I interpreted your posts as asking for. One key point though is this image allows to extrapolate 2d relations. I assume your looking for a 3d based on these relations with a different sphere reference. Why would you think it requires a function??? https://www.mathsisfun.com/definitions/function.html

This seems to be the gist of what your after but I'm still not clear on your understanding of vectors in particular your two dimensional section. A vector in three dimensions can be expressed simply as P_1=(x_1,y_2,z_3) with end point P_2(x_1,y_2,z_3) Simply draw a line between those two points and there is your vector. However I know this doesn't help for what your trying to do. Probably because I don't particularly see the need for the above. Let's explain we will use vectors for now. In 3D [latex]\mathbb{R}^3[/latex] U is the set of vectors in [latex]\mathbb{R}^3[/latex] where[latex] (x,y,z)\in\mathbb{R}^3[/latex] The four basis vectors being [latex]e_1=(1,0,0).. e_2=( 0,1,0).. e_3=(0,0,1).. e_4=(0,0,0)[/latex] Then using [latex]P(x,y,z),\acute{P}(\acute{x},\acute{y},\acute{x})[/latex] Then [latex]u=(\acute{x}-x,\acute{y}-y,\acute{z}-z)[/latex]

Ok well this is definitely a Smells law project. So in that regard you have pieces of information outside what you posted. First is the law of reflection. [latex]\theta_I=\theta_r[/latex] The next is the index of refraction. For air the index of refraction is 1.00. So on your table I assume you are given different index of refraction to work with? You give one value n_1 being air but only provided one medium index of refraction for the prism being n_2 equals 1.5 Noted your image didn't copy on the site however you can type in the values you are given for each row and type in the columns you need to fill. Note you'll be using the law of refraction. This law can be stated in two parts. 1) [latex]n_1*sine\theta_1=???[/latex] 2) what is the relation between [latex]\theta_1 and \theta_2 [/latex] I didn't provide the answers to those two parts as I want you to fill in the blanks. The first part should give you the the detail you need to calculate for your second angle, keeping in mind the second part.

Truthfully there comes a point where you need to attempt the math yourself. It's extremely difficult to develop a math for someone's else's idea when they themselves don't make any effort to post their own math. For one fact there is no way to judge what your math skills are to develop a mathematical model that you may or may not understand. Secondly it may help better explain what you are attempting.

None that I recall.

True but I don't think the OP would distinquish the difference. It takes a good working knowledge to understand how dot products work. Yes it's easy to apply the dot product between two vectors. Hrrm I'm not sure if I can latex dot products lol. Here goes. [latex]a \cdot b=|a|*|b|*cos(\theta)[/latex] Hmm not quite right I don't want the dot on top of b but between a and b. Ah there you use \cdot. (Thank god for online reference links lol. This time wiki lol) Locally though manifolds resemble Euclidean space.

I can't even imagine describing a point in space without using some form of coordinate. At least not in any mathematics.

Again you still have the problem. An arrow is easily described by the vector equation above. So you have direction. The only difference is you also have a magnitude. Direction is always relative to a reference. One can easily state go 20 degrees from point P. But then everyone has to agree that 0 degrees follows the positive y axis... You can have as many coordinates as you want but you still need a reference point. The above math uses 0,0,0 as the reference. This is the point you need to specify. You cannot describe direction without a reference. Angles from a starting coordinate is easily quantifying a direction but you have a coordinate system. The short answer is you need a reference to quantify a direction. It doesn't work to say go left if you don't know which way your facing. For example going upstream is a direction one that is against the current flow.

You had better reword this last post in relation to the OP. In particular the thread title. I don't like wasting my time on confusion. The majority of your thread has been describing direction. In one example you specifically asked for tensors and vector relations. You want to describe direction, as a function. I already explained as well as others that direction isn't invariant. Vectors have both magnitude and direction. In this there is plenty of examples. Yet you choose to ignore why the scalar and not the direction quantity is invariant. The math I'm posting shows why this is the case. In the last post I made you can apply your rotations for 45 degrees in precisely the same manner as the link you provided. Yet you obviously don't want those details. Essentially what I've done above is shown A) how to describe a vector and b) the rotation rules in 2d. You can Google the 3d and 4d rotation rules.

If you work it out I have been. You wanted to understand how to mathematically describe a vector as a tensor. So in order to explain we needed to determine what units will remain unchanged regardless of translational or rotational change. That value is the vector length. Which is the magnitude. Yes it's easy to describe a vector using Cartesian coordinates, Take an x,y graph. Using coordinate 0,0 as a start point then describe the opposite coordinates. You can extend this by adding the z component and for 4d use w. But all that is reference based. For example a vector with origin point 0,0,0 can be described by [latex] a=ax I+ay j+az k[/latex] [latex]a=\begin{pmatrix}a_x\\a_y\\a_z\end{pmatrix}[/latex] Now I just described in matrix form a vector. Did that somehow help if you don't understand how matrix and tensors work with coordinates i,j,k '(unit vectors)or in Einstien summations? Yes I described a vector, I can apply translation or rotations to this but the rules will vary according to those described by tensors. For example a 2d rotation of a line with 0,0 as the starting point will look like this in matrix form [latex]\begin{pmatrix}\acute{x}\\\acute{y}\end{pmatrix}\begin{pmatrix}\cos\theta&-sine\theta\\sine\theta&cos\theta\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}[/latex] The last set is your rotation

In all honesty the closest you will get is to study how to represent vectors in tensor form. However the tensor translations will have different coordinate transformation rules. The links I provided on tensors are some examples. You can easily Google more on vector calculus and by googling vector to tensor transformations. Type pdf at the end (it will give a better ratio of well written articles) These will guide you into mathematically describing a vector to perform a rotation or translational symmetry transformation. You can do a simple example yourself. Draw a line on a graph at a diagonal. Then take each point on the line and count three units right and 3 units up. When you connect those points you will have a vector with is identical to the original just translated to a new location. Mathematically describing this is covered in the links provided.

Perhaps you better look at the math definition of a scalar vs a vector quantity. Particularly if your going to post incorrect claims.. Scalar. (of a quantity) having only magnitude, not direction. Precisely note the need for a reference. Needing a reference is not invariant Anyone can describe direction to a reference. Everyone does so... No one describes a direction without a reference.

Well now your starting to think. What does the term movement imply? What physical laws are involved to determine movement? So thinking of the above in regards to time dilation. What causes an object to move slower in time compared to another time frame? Ie f=ma. You beginning to see the hazard of stating objects move through time particularly since time isn't invariant? In the expansion example. Take a homogeneous and isotropic fluid and surround every galaxy. At no point is there a higher pressure or force acting on any galaxy in any direction. Yet galaxies still move away from each other. No galaxy gains inertia that requires a pressure difference on a facing. Yet they all appear to be accelerating apart. We describe this simply as geometry change. Same principles apply when describing an object moving in time. You have different measurements of the passing of time on the same object. How do you describe that as movement? It's easy to describe as geometry change.

Oh which geometry is direction a scalar? Go ahead name it.

Look your still missing the point. Different observers will measure the direction differently. Its always change relative to a reference. That is not an invariant property. Hence direction is a relationship, in the same manner as distance. For example the direction up is relative to your choice of down. (Normally chosen by which way objects normally free fall) A person on the other side of the planet his up is different than your up direction. How you would describe North on Uranus would be different than on Earth. You can draw any line on a circle and state this is zero degrees regardless of orientation of that circle.

Well tell you what when you show the mathematical impossibility of describing a direction as being invariant to all observers let me know. You can't even show that as being invariant in the same coordinate system. Observer a measures a ball moving in one direction (relative to himself) another observer will state its direction relative to himself as different. That by itself is inherently variant to the observer Even certain measured properties are variant. One example being the measured energy or temperature. No amount of mathematics will change this. It is inherently part of relativity. Google redshift for details It isn't some arbitrary mathematical trick that defines whether or not something is invariant. We would have loved time to be invariant. However tests of relativity shows that as being false. It certainly would have simplified things if light was variant and time invariant. Nature doesn't care about what we wish though. PS hopefully you don't post these speculations on physicsforum. They will instantly lock the thread at any hint of speculation. Yes I'm a member there as well

That's my point there is no math that will make direction invariant to all observers regardless of coordinates. If I could show that mathematically I would win the Nobel prize. Look at the examples I posted. Different observers will measure different directions How they describe that direction is a coordinate choice. What they compare that direction to is also an arbitrary choice. Arbitrary coordinate and reference points are not invariant.

Good luck on making direction invariant. That's a handwave if I ever saw one.

Is the object moving or are the coordinates changing ?. What laws are involved when considering inertia? Implying that objects move through time implies falsely that time acts as a force. A good example of the types of misconceptions is say for example spacetime expansion. Which doesn't impart momentum to galaxies. No object gains momentum due to expansion. No object gains momentum due to different time rates. Remember the laws of physics including momentum must be the same for all observers. For example the four vector length is invariant. http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/vec4.html When you apply a Lorentz boost your applying a coordinate transformation. That coordinate transformation does not change the laws of physics in regards to momentum. This is why the terms proper time and coordinate time is also important. https://en.m.wikipedia.org/wiki/Coordinate_time For example I am an observer in the same reference frame as a reference clock. There is no time dilation. However another observer with momentum measures that same clock and sees a time dilation. Mathematically we treat time as a coordinate with the speed of light being invariant to account for these two measured rates of time.