1. Sub Quantum Echo Particles...(SQEP's) & Sub Quantum Echo Particle Kinetic Resonance Flux

[studiot]

15 hours ago, Imagine Everything said:

 

Thanks also Studiot, I'll have a read when I have more time later on.

 

Might I make a small suggestion ?

 

On 11/12/2024 at 8:20 PM, Imagine Everything said:

My head likes to see things in images, though I think I'm good with english, it's much easier for me to see these things in action as images/animations with their definitions & explanations

 

Both Mordred and I try to develop a train of thought (or chain of ideas) when we write something down.

If you tried to draw this as a diagram perhaps it would help.

For instance the names etc ofthe shells and subshells have been listed several times.

 

If you went through the list and as you read each idea you wrote it down it might look something like this at the end.

 

It doesn't matter if you just scribble it down, or if you post it or not,  it is for you not us.

Edited November 20 by studiot

[Imagine Everything]

14 hours ago, studiot said:

 

Might I make a small suggestion ?

 

 

Both Mordred and I try to develop a train of thought (or chain of ideas) when we write something down.

If you tried to draw this as a diagram perhaps it would help.

For instance the names etc ofthe shells and subshells have been listed several times.

 

If you went through the list and as you read each idea you wrote it down it might look something like this at the end.

 

It doesn't matter if you just scribble it down, or if you post it or not,  it is for you not us.

lol, I feel a bit  silly now, yes yes yes ofcourse you can and why didn't I think of this already....

I focus too much sometimes on complex things that I forget the simple things.

Edited November 21 by Imagine Everything

[Mordred]

One of lifes rules for every trade, science, programming etc. All complex problems are composed of simpler problems. 

lol or whats that expession one bite of the apple at a time.

[Imagine Everything]

1 hour ago, Mordred said:

One of lifes rules for every trade, science, programming etc. All complex problems are composed of simpler problems. 

lol or whats that expession one bite of the apple at a time.

My friend you are not wrong.

I've just been watching a couple of lectures on spin and symmetry. Read a little about pions earier too.

 

I take back what I said before, maybe it'll take a good few years to learn about electron orbits / clouds / spins 😮 

So much to learn...oh well lol    "here i go again on my own, going down the road I know I need to learn"

My own 'spin' on a cool song

Edited November 21 by Imagine Everything

[Mordred]

Lets tackle symmetry before spin as symmetry can be easily shown and I will work up an easy way of understanding symmetry

[Imagine Everything]

Ok, throw it at me lol, lets see what I can learn and hopefully remember.

Edited November 21 by Imagine Everything

[Mordred]

7 hours ago, Imagine Everything said:

Ok, throw it at me lol, lets see what I can learn and hopefully remember.

lol you can relax on this one symmetry is in actuality easily understood its learning which mathematical relations are symmetric that tends to get confusion.

lets first take a simple everyday example. One of my favorite examples is to use fan blades.. Each blade is identical so they are symmetric to each other.

As the fan rotates those blades are unaltered this is called rotational symmetry.

if you move the fan around the room the blades remain unaltered. this is called translational symmetry.

if you look at an image of the blades in a mirror they look identical (reflection symmetry)

if you combine both translation symmetry and reflection symmetry this is called glide symmetry.

Now on the mathematics side.

if you add a set of numbers or values and the order of operation doesn't change the resultant the equation is commutative

 https://en.wikipedia.org/wiki/Commutative_property

if an equation is commutative then its relations are symmetric.

so for example with states this can be expressed as

\[|\psi\rangle_{c}=|\psi_a\rangle +|\psi_b\rangle\]

the above tells us the order you add the states doesn't matter. So it is a symmetric relation

if on the other hand you have

\[|\psi\rangle_{c}=|\psi_a\rangle -|\psi_b\rangle\]

the order of operations does change the result this is an antisymmetric relation.

it is common nomenclature to place the negative sign as the first term on the RHS of the equal sign

\[|\psi\rangle_{c}=|-\psi_a\rangle +|\psi_b\rangle\]

multiplication is commutative (symmetric) however division is not (antisymmetric)

Now consider spin. Boson spin is an integer 1 most common zero in Higgs.

You can add or subtract bosons in any sequence and the order of operation doesn't but also other than total energy and number of bosons in the system no other change occurs as a result.

This isn't true for fermions with half integer spins ie 1/2 the order of operations does matter . It is also this detail that leads to the Pauli exclusion principle.

https://en.wikipedia.org/wiki/Pauli_exclusion_principle

In bosonic systems, all wavefunction must be symmetric under particle exchange.

for bosons

\[\psi(x_1,x_2)=\psi(x_2,x_1)\]

the above is a commutative expression (symmetric)

for fermions

\[\psi(x_1,x_2)=-\psi(x_2,x_1)\]

this is an anticommutative expression (antisymmetric)

Last two expressions are called Slater determinants

https://en.wikipedia.org/wiki/Slater_determinant

the matrix in that link can readily be understood here

https://www.khanacademy.org/math/multivariable-calculus/thinking-about-multivariable-function/x786f2022:vectors-and-matrices/a/determinants-mvc

that link has an interactive graph you can play around with ( Pay close attention to how the parallelogram changes shape as you interact with it this will become important to understand spin later on.)

but also a link to 3d Determinants

https://www.khanacademy.org/math/linear-algebra/matrix-transformations/inverse-of-matrices/v/linear-algebra-3x3-determinant

This should also help better understand that with vector addition the inner product of two vectors

\[\vec{a}\cdot \vec{b}=\vec{b}\cdot \vec{a}\]

is symmetric (commutes) the order of inner product of two vectors doesn't matter

where as the cross product of two vectors anticommute 

\[\vec{a}\times\vec{b}=-\vec{b}\times \vec{a}\]

now some instant recognition rules with matrix/tensors

\[\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}\]

this matrix is symmetric and orthogonal 

https://en.wikipedia.org/wiki/Orthogonality

the only non vanishing terms are on the diagonal given as 1 in each entry any non vanishing term not on the diagonal above is antisymmetric

For QM if all diagonal terms are a real number (set or reals) that matrix is a also Hermitean

https://en.wikipedia.org/wiki/Hermitian_matrix

as you can see in the above a Hermitean matrix need not be symmetric but all diagonal terms must be a real number.

this will help when it comes to spin statistics under QM example a preliminary lesson to understand 

https://en.wikipedia.org/wiki/Conjugate_transpose

but before we deal with the complex conjugate under QM treatments lets familiarize you with what a complex number is.

as You learn best from videos 

https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex/x2ec2f6f830c9fb89:complex-num/v/complex-number-intro

now recall that interactive video where I asked you to pay attention to the parallelogram. ?

replace the two axis as one axis imaginary numbers the other axis Real numbers. Just like the complex number wiki link images and make further note that the parallelogram in the interactive map  is identical to the parallelogram in that same link and that we can add two complex numbers using the parallelogram.

 this will become important later on when we assign operators as those complex numbers as well. (under QM two spinors which are also complex number for what is called a bi-spinor giving us 4 complex numbers in total needed for Dirac matrices later on). This will also apply to linearization of non linear systems. Take your time on this as it will become incredibly useful for a great many different physics treatments including GR/QFT etc. @studiot will also recognize the above applies to classical physics as well as engineering applications.

further consideration as your looking at the parallelogram I want you to also consider the following 

(eigenvectors and eigenvalues)

https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

In linear algebra, an eigenvector (/ˈaɪɡən-/ EYE-gən-) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation.

if you look at the interactive graph on that last wiki link you can see the connection. to the Khan University interative map.

the below will also help for those last two terms

https://www.mathsisfun.com/algebra/eigenvalue.html

note from last link

• 1 means no change,
• 2 means doubling in length,
• −1 means pointing backwards along the eigenvector's direction

 

now look back at the values in top left corner of the first khan interactive map on determinants. The box with the changing numerical values in matrix form

(2 by 2 matrix)

All the above is called skew symmetry. As you stretch the eugenvector the parellelogram becomes scewed/stressed.

(will apply to all energy momentum stress tensors). Further a skew symmetric geometry can be Hermitean (assign real values only on the eugenvector) while the geometry is not orthogonal (its now skewed)

(you may also want to notice that the interactive determinant graph is showing rotation and reflection symmetry ie when i axis is in line with j axis. They have reflection symmetry with one another. Vectors where the only change is direction have rotational symmetry.

Edited November 22 by Mordred

[Mordred]

 

 the interactive parallelogram  for determinant's  (khan link above we can assign it as a matrix any assignment will do so lets just call it matrix B so that matrix is the box with two rows and two columns. The rows and columns are equal so it is a square matrix.  Adding the \(\hat{i},\hat{j}\) coordinates we can place those coordinates in the (subscript= bottom, superscript=top exponent).

\(B_{i,j}\) on subscript, \(B^{i,j}\) where " i " is the row vector," j" is the column vector.

Now recall those spin states.  Each spin has a limited set of allowed values for the S component.

for spin 1/2 particles 

https://en.wikipedia.org/wiki/Doublet_state

In quantum mechanics, a doublet is a composite quantum state of a system with an effective spin of 1/2, such that there are two allowed values of the spin component, −1/2 and +1/2. Quantum systems with two possible states are sometimes called two-level systems. Essentially all occurrences of doublets in nature arise from rotational symmetry; spin 1/2 is associated with the fundamental representation of the Lie group SU(2).

for Spin 1 bosons.

https://en.wikipedia.org/wiki/Triplet_state

 

In quantum mechanics, a triplet state, or spin triplet, is the quantum state of an object such as an electron, atom, or molecule, having a quantum spin S = 1. It has three allowed values of the spin's projection along a given axis m_S = −1, 0, or +1, giving the name "triplet".

Spin, in the context of quantum mechanics, is not a mechanical rotation but a more abstract concept that characterizes a particle's intrinsic angular momentum

 

hopefully the above helps as a means to see what the math is doing without knowing the math. Going to let you study the above as there is a lot to absorb on the last post

https://en.wikipedia.org/wiki/Spin_quantum_number

In the last link look at the reflection symmetry for spin up and spin down.

 

 

 

 

Edited November 22 by Mordred

[Imagine Everything]

In the Slater determinant are the x1 & x2 the electrons in the 1st part before the =?

I'm trying hard to understand this better but hmm, is the 2nd part of it after =, simply the wavelength without the wavelength symbol in front of it?

Does the commar in the first x1,x2 have a relevance other than splitting the 2 x's up?

This is as far I have got so far, still reading, still mostly confused but it's a lot to take in all at once, so I don't expect I will make much sense of it atm. I will eventually I hope...

Oh and when I see RHS mentioned, does it mean Right Hand Side?

 

Going by the Pauli exclusion about Fermions, how is it that a nucleai can have the same protons in the same space, or is this not what is going on in the nucleus?

 

Is it rather, that the protons are simply close to each other in a tight space?

Perhaps I've read too much today, the determinants are starting to drive me nuts a bit too.

I'll try again tomorrow.

 

Edited November 22 by Imagine Everything

[Mordred]

The x is a placeholder for any relation being compared example a vector or spinor or even some geometry or object. Yes the comma is just seperation for legibility.

Protons are simply tightly packed and Pauli exclusion applies to protons as well.

RHS is right hand side

In those graphs think of the determinant as the volume distortion but it can have different meaning in other systems or more accurately how much the linear transformation will affect the volume

Edited Friday at 11:39 PM by Mordred

[Imagine Everything]

8 minutes ago, Mordred said:

The x is a placeholder for any relation being compared example a vector or spinor or even some geometry or object. Yes the comma is just seperation for legibility.

Protons are simply tightly packed and Pauli exclusion applies to protons as well.

RHS is right hand side

Thanks, just curious, did you Studiot or Swansont or even any other scientist / mathematician get frustrated when you started learning about maths and science?

[Mordred]

1 hour ago, Imagine Everything said:

Thanks, just curious, did you Studiot or Swansont or even any other scientist / mathematician get frustrated when you started learning about maths and science?

All the time that's usually when I usually take a mental break and sit back to figure out what detail I'm missing.

It's very common, the trick is to not stress about it and go back and few steps to look for the missing details or look for other references that may have a different writing style, examples or different math treatments and look for the elements common to both treatments.

There are various equations common to all physics example the energy momentum relation, scalar, vectors and spinors 

This is also why I keep dropping reminders to learn vectors and vector addition. Every physics theory involves them and once you master vector particularly in vector field treatment the vast majority of physics becomes far easier to understand regardless of theory.

Example spin is angular momentum so I automatically know I will need cross products of vectors.

Linear relations only requires the inner product of vectors.

Conserved systems are systems where no force is applied example freefall condition under GR or Newtons laws of inertia. 

An object will maintain a constant velocity until acted upon by a force=acceleration. Acceleration is not conserved but velocity is.

For another example if no force acts upon a particle you can describe the particle as a conserved system. When a force acts upon that particle its no longer conserved.

Invariant properties (same to all observers are conserved systems )

All conserved systems are closed as you are describing the system or state without any outside influence. Closed systems must also be finite in extent.

The inner product of 2 vectors returns a scalar how convenient we now have the magnitude.

The cross product of two vectors returns a new vector so now we have the change in direction. How convenient.

It's so convenient it's common to all physics theories. 

Hence also why I pointed out similarities that  QM has with classical physics. Classical physics provides a solid foundation to understand more complex physics.

If you ever want to understand how it is I can make sense out the most complicated physics treatments the above describes precisely how I so.

  I find where classical physics can be applied  and then study where more complex examinations becomes required and why they are required.

 

 

Edited Saturday at 01:37 AM by Mordred

[Imagine Everything]

On 11/22/2024 at 1:22 AM, Mordred said:

https://www.khanacademy.org/math/multivariable-calculus/thinking-about-multivariable-function/x786f2022:vectors-and-matrices/a/determinants-mvc

Thanks, I took a day out from it all or so I thought.

The above link, about halfway down shows a 3d determinat matrix (is matrix the right word)

and I am being frustrated by their answers to the 3rd determinant

I'm not asking for answer, I need to figure it out but thats where I am atm.

So I thought I took a day out, hahaha my bloody brain wouldn't shut up.....so

On 11/22/2024 at 11:44 PM, Mordred said:

Conserved systems are systems where no force is applied example freefall condition under GR or Newtons laws of inertia. 

An object will maintain a constant velocity until acted upon by a force=acceleration. Acceleration is not conserved but velocity is.

For another example if no force acts upon a particle you can describe the particle as a conserved system. When a force acts upon that particle its no longer conserved.

Invariant properties (same to all observers are conserved systems )

All conserved systems are closed as you are describing the system or state without any outside influence. Closed systems must also be finite in extent.

Not too sure about the underlined part, still need to learn more.

By the bit I emboldened wouldn't leave me alone.

I  make the mistake now and then of thinking of literal particles and not energy/wave lengths.

I this 'thingy' in my idea exists, from what I have learnt and if it indeed is created to die instantly, it would have to exist as energy I think. I'm still learning as you know so please bear with me.

So going back to the emboldened part, it kept making me think and think and think, what could it be if it is in an enclosed/conserved state / system...hmmm...and more hmm..and more hmm..

Then it struck me, perhaps I'm only thinking or have only suggested this in 2d.

Atm I can't think of anything that would only have sides. It has to have surfaces too (3d) and therefore state 3 would perhpaps be the meeting of the 2 boundary condtions but maybe..just maybe, there's a bit of quantum tunneling going on and it might not be a 'totally' conserved state even though it is (mmm it sounds weird to me too) due to quantum tunneling (Im still learning about that too).

The surface of both state 1 and state 2 would also be next to something making this perhaps a 3d kind of state.

After all, isn't everything, even if it is part of a chain (all be it a universally sized chain, in all directions) be connected to each other. Whether 1st gen connection (direct contact), 2nd gen connection, or even 1 billionth gen connection ...

So whatever the surface/s is/are in contact with are somehow helping to create this thingy in state 3.

A question: Do nuclear force & weak force not count in a conserved system?

I'll leave it there as it is already confusing me and thoug I think I can sort of see it in my head, I don't know enough to present it in written or maths form.

You guys know more than I do and idk how stupid that may or may not sound to you, but this is kind of where I am so far.

Off to try and understand 3 det now again...

Edited Sunday at 01:54 PM by Imagine Everything

[Mordred]

Yes the 4 entry box is a matrix. That matrix has entries those entries don't worry about just yet but they are derived by the inner products of the i,j vectors. ( part of the linear algebra that the link doesn't show so  don't fret of the entries itself as I have taught you how to fill those entries 

for the determinant you are performing a math exercise using the entries 

\[\begin{pmatrix} a&b\\c&d\end{pmatrix}\]

so lets set the entry values a=2 b=3 c=4 d=5

the formula for determinant is

\[ad-bc\]

so the determinant is

(2* 5)-(3*4)\] so 10-12=-2

which if you look at the descriptive's in the link if the sign changes from a positive to negative determinant  the orientation of the parallelogram is flipped over but also the value (2) is greater than 1 so it is stretched out.

that is the general purpose of the determinant its to provide a  means of scaling For quantum spin the main focus is the sign reversal itself specifically spin up and spin down. The entries for that matrix in the third link will be using a specific set of linear equations that will be determinant from linearizing the spin angular momentum operators. Which we haven't covered yet.

[Imagine Everything]

I've had to back track a lot on Khan academy to linear vectors and real co-ordinate spaces, I was getting very lost seeing 4 + -1 = 5 - still not sure why that isn't 3.

Anyway, it's talking about more than 3d which is interesting.

I guess I'm writing this because as I said in my first post, this is a very simple idea but can get very complicated.

Little did I know what I was talking about even though I was talking about it lol

Anyway, perhaps my 'simple idea' is actually more like 4d, 5d or even further along. I don't know if thats even possible or how far it goes if it is.

Left side/right side

Inside/Outside

Surface/top/bottom

Diagonals

Nuecleus/decay

Information transfer? (quantum hair)

Boundary condtions & the behaviour of electrons?,  between the boundary conditions, perhaps also what is happening within the states themselves and how they may be affected by quantum tunneling? (if thats a possibility, I saw this as  leakage before you mentioned quantum tunnelling to me), fields (EM/Higgs?) and idon't know what else.

I have a lot of lectures to watch now before I get back to the determinant lecture you gave me a link for Mordred.

 

 

 

4 minutes ago, Mordred said:

Yes the 4 entry box is a matrix. That matrix has entries those entries don't worry about just yet but they are derived by the inner products of the i,j vectors. ( part of the linear algebra that the link doesn't show so  don't fret of the entries itself as I have taught you how to fill those entries 

for the determinant you are performing a math exercise using the entries 

 

(acbd)

 

so lets set the entry values a=2 b=3 c=4 d=5

the formula for determinant is

 

ad−bc

 

so the determinant is

(2* 5)-(3*4)\] so 10-12=-2

which if you look at the descriptive's in the link if the sign changes from a positive to negative determinant  the orientation of the parallelogram is flipped over but also the value (2) is greater than 1 so it is stretched out.

that is the general purpose of the determinant its to provide a  means of scaling For quantum spin the main focus is the sign reversal itself specifically spin up and spin down. The entries for that matrix in the third link will be using a specific set of linear equations that will be determinant from linearizing the spin angular momentum operators. Which we haven't covered yet.

I undertstood this, a*d - b*c =

But when it went 3d..well thats where it went a bit confusing for me.

[Mordred]

1 hour ago, Imagine Everything said:

 

I undertstood this, a*d - b*c =

But when it went 3d..well thats where it went a bit confusing for me.

that is understandable it gave me trouble as well when I first learned this but if you figured out the 2d case your on the right track and doing well. Just a side note that plane your moving around is a complex plane and you may have noticed you can orient it in 3d. The entries for i and j had previous math operations done to it for the linearization they are complex conjugates denoted by the 

\[\hat{i}\] hat on top

when it comes to quantum spin we will have to calculate these as the vertical will be imaginary numbers and the horizontal will be real numbers. ( we will need those for charge conjugation ie CPT charge, parity and time symmetry) later on

hint your boundaries is the edges of the parallelogram  for spin

its the edges of the possible values ie the edges of the shaded area of the spin graph

https://en.wikipedia.org/wiki/Spin_quantum_number

the shaded area on that graph are the range of possible values ie the quantization which we have to cover specifically the math in this link

https://en.wikipedia.org/wiki/Azimuthal_quantum_number

hopefully though this will take considerable time we can get you to understand this link

https://en.wikipedia.org/wiki/Angular_momentum_diagrams_(quantum_mechanics)#Inner_product

but we need to sharpen your classical physics before taking on the challenges of QM

 

Edited Sunday at 05:03 PM by Mordred

[Imagine Everything]

My head is a bit foggy again with magnitudes, vector units, square roots and things.

Bit confusing right now but I think I understand vector and scalar more now so thats something.

Have a good night. Thanks for your help

 

[Mordred]

night and its great your starting to understand scalars and vectors keep at it.

Here is an assistant article just on  units and vectors for physicists

https://www2.tntech.edu/leap/murdock/books/v1chap1.pdf

in particular

\[A\times B=\begin{pmatrix}i&j&k\\a_x&a_y&a_z\\b_x&b_y&b_z\end{pmatrix}\]

which in other notation equals

\[A\times B=(a_yb_z-a_zb_y)i+(a_zb_x-a_x b_z)J+(a_xb_y-a_yb_x)k\]

this will correspond to that right hand rule I mentioned quite a while back also those matrix you have been looking at for the 2*2 matrix drop the k terms.

You can project a 2d plane in any orientation in 3d. The 3*3 case is two complex planes where as the first case is one complex plane (first case SU(2) second case SU(3). Those are two fundamental particle physics groups which is where this lesson becomes valuable. For spin you need the SU(2) gauge group those vectors above are applied under all unitary groups U(1), SU(2),SU(3). for higher dimensions we will need higher vector commutations.

we can only fill a matrix entry with scalar values so we need to get scalar values from vectors. That is why this lesson is so important.

By the way once you understand the above you will have the necessary tools to understand Special Relativity and the Minkowskii metric. 

We can step into SR quite readily once you are comfortable with the above.

Were literally on the edge of stepping you into electromagnetism via Maxwell equations 

 

Edited Sunday at 06:36 PM by Mordred

[studiot]

Just now, Imagine Everything said:

I've had to back track a lot on Khan academy to linear vectors and real co-ordinate spaces, I was getting very lost seeing 4 + -1 = 5 - still not sure why that isn't 3.

Diagrams, Diagrams, Diagrams.

Does this sketch help ?

 

[Mordred]

9 minutes ago, studiot said:

Diagrams, Diagrams, Diagrams.

 

 

lol greatest training aid graph paper

[studiot]

Since you are discussing vector based quantum mechanics with mordred you will find this page invaluable.

You should print it out for reference.

Note how simple they are.

 

 

[Mordred]

This is more for fun but it also demonstrates another aid. Lets say you have a 2D vector field. We can graph this vector field as a function for simplicity lets do a 2D center of mass. I happen to know the function for this 

\[F(x,y)=(-x,-y)\]

now here is a neat trick there is a handy tool Wolframalpha.com

its entries can be tricky but the above is

https://www.wolframalpha.com/input?i2d=true&i=plot+F\(40)x\(44)y\(41)%3D<-x\(44)-y>

now lets say we want the curl of a magnetic field in 2D.

\[F(x,y)=(-y,x)\] 

https://www.wolframalpha.com/input?i2d=true&i=plot+F\(40)x\(44)y\(41)%3D<y\(44)-x>

this is just to demonstrate the usefulness of functions 

Here is what Function of sin(x) looks like see link

https://www.wolframalpha.com/input?i2d=true&i=plot+sin\(40)x\(41)

here is cos(x)

https://www.wolframalpha.com/input?i2d=true&i=plot+cos\(40)x\(41)

Now you can do the same with tan(x) and it will look completely different. However the main goal is to point out the WolframAlpha has some handy plotting capabilities as well as numerous others to help check your math or get a better feel for some of the more complex equations.

The first graph is a converging field example gravity converging to a center of mass If I switch the sign of x  I will type this in as WolframAlpha wants the format

Plot F(x,y)=<( -y,x)>

the field will be diverging from the center outward. Just copy and paste that line and it should work.

Graph 2 is an example of a non diverging curl (rotationally symmetric)

good example application is the magnetic field as opposed to the electric field which will be this example. I will let you enter that. However that depends on charge the opposite charge will be example 1.

Plot F(x,y)=<( -y,x)>

Hope that helps better visualize what functions do in terms of fields

One particular handy use will come into play to better understand matrix mathematic for example

\[\begin{pmatrix}a & b\\c&d \end{pmatrix}*\begin{pmatrix}e&f\\g&h \end{pmatrix}\]

https://www.wolframalpha.com/input?i=[[a%2Cb]%2C[c%2Cd]]*[[e%2Cf]%2C[g%2Ch]]

this will solve that operation for you and give a step by step (though you get far greater detail if you pay for membership) the details it does supply to non members is often as useful.

Just an additional training aid to help you along

 

 

Edited Monday at 03:55 AM by Mordred

[Imagine Everything]

12 hours ago, studiot said:

Diagrams, Diagrams, Diagrams.

Does this sketch help ?

 

Immensely, a kind of Rosetta stone ty.

 

 

Edited Monday at 07:29 AM by Imagine Everything

[studiot]

Since you and Mordred are ploughing on with the advanced stuff here are some simple notes about scalars vectors and

tensors.

Again I sugest yoy print the attachment out and keep it for future reference.

Sorry it's all words this time but the best introduction I know without using the Einstein summation notation.All the essential features are there.

 

First a couple of notes of my own.

17 hours ago, Imagine Everything said:

linear vectors

All vectors are linear.

Mostly when a physicist says vector he means a 'directed line segment' or a 'signed line segment.'

note in maths a line is a straight line that goes on to infinity or forever in both directions.

That is it has no ends.

A 'straight line with ends ' is called a line segment.

A 'curved line' is called a curve and a piece of that curved is called a curved segment or a segment of the curve.

In my previous sketch about -1 + 4 = 5 I have shown three line segments.

One property of any segement, is that it has length.

So if I add two segments that each have a length the result is a longer segment of length equal to the sum of both individual lengths.

This form of addition is different from the rules for adding signed  numbers.

This is why you have to consider the meaning of the sign as well as the magnitude (length in this case).

 

@Mordred

Thie following is from Physics in the Chemical Industry by R C L Bosworth  - Macmillan.
A no nonsense guide as it says.

 

 

Edited Monday at 03:00 PM by studiot

[Imagine Everything]

Thanks both of you, still watching the lectures atm, don't know nearly enough yet.