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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. 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.

What do you mean "she wasn't chosen"? She was elected as VP, voted into office by the People. What's the role of the VP, you might ask? One part is to take over when the president can't fulfill the duties. And Biden left us one of the strongest economies in recent history. So strong it encouraged US corporations to price gouge their own customers, knowing the uneducated would blame Biden, as if POTUS has control over private companies and what they charge. And yet you don't bother to link to it here. I'm going to assume you made it up. OTOH, TFG has decades of ruining people's lives under his belt: https://www.cnn.com/2016/09/13/politics/trump-small-business-owners/index.html https://www.northjersey.com/story/news/columnists/mike-kelly/2020/01/24/donald-trump-still-owes-money-to-contractors-who-built-taj-mahal-atlantic-city/4547037002/ https://nj1015.com/trump-stiffed-hundreds-of-workers-and-contractors-in-new-jersey-report-says/ https://theweek.com/articles/783976/brief-history-trumps-smalltime-swindles https://www.nbcnews.com/news/us-news/hundreds-claim-donald-trump-doesn-t-pay-his-bills-n589261 He's a swindler, so I assume you cheat people too. None of this is a deal breaker for you. People like you and him kill thousands of decent Americans every year. He's a racist, so I assume you hate people of color too. None of the bigotry and prejudice bothers you enough to make it a deal breaker. People like you and him discriminate against the only known humans in the universe, and ignore the diversity of life we have on this planet. I hope you don't pass this behavior to the next generation. He's fooled you because of all your ignorance and hatred. You're the poorly educated people he truly loves. Harris kept trying to tell you what she was going to do for us. TFG keeps telling you who he's going to punish, and you evangelicals just lap it up. It's all about punishment and atonement and how shitty humans naturally are, and now you have the shittiest human on the planet as your leader. Get down on your knees and take him there!

Did you know that Nixon's strategists brought the Southern white vote over to the Republican party through their opposition to Johnson's desegregation policies? I don't suppose you did. It was already working for Goldwater, and it's continued to work right up to the present. Over time and election molesting by state legislations, it expanded to include the anti-immigrant factions, the religious right, misogynists, anti-intellectuals, gun nuts and haters-at-large. All Trump's people had to do was dip into that deep dark well of carefully nurtured bigotry. They don't care what he's done or what he will do, as long as he socks it to whomever they most fear. In this instance, independent women topped the list.

In special relativity, the energy-momentum vector is a four-component vector in spacetime. The time component corresponds to the energy, the three spatial components correspond to the momentum, and the magnitude of the vector corresponds to the invariant rest mass. (Note however that for dimensional consistency of units, E=pc=mc2). Because it's being described in the earth's frame of reference, and in that frame of reference, the energy component of the four-component energy-momentum vector is increasing, though the invariant magnitude of this vector is not changing. The formula that describes the invariant rest mass in terms of energy and momentum is: [math](mc^2)^2 = E^2 - (pc)^2[/math] ........................................ [math](m_{\text{rest}}\ c^2)^2 = E^2 - (pc)^2 \\ E = m_{\text{relativistic}}\ c^2 \\ p = m_{\text{relativistic}}\ v \\ (m_{\text{rest}}\ c^2)^2 = (m_{\text{relativistic}}\ c^2)^2 - (m_{\text{relativistic}}\ \dfrac{v}{c}\ c^2)^2 \\ m_{\text{rest}}^2 = m_{\text{relativistic}}^2\ \left(1 - \dfrac{v^2}{c^2}\right) \\ m_{\text{rest}} = m_{\text{relativistic}}\ \sqrt{1 - \dfrac{v^2}{c^2}} \\ m_{\text{relativistic}} = \dfrac{m_{\text{rest}}}{\sqrt{1 - \dfrac{v^2}{c^2}}} [/math]

The assumption is that the Falcon 2 is getting its power to accelerate from an external source, and that the rest mass of the Falcon 2 is not changing.

No. If an object is at rest relative to the Falcon 2, then its mass in the frame of reference of the Falcon 2 will be the rest mass of the object. Note also that in the frame of reference of the Falcon 2, the mass of the Falcon 2 will be the rest mass of the Falcon 2. Also, in the frame of reference of the Falcon 2, the mass of the earth, which is moving at 99.99%c relative to the Falcon 2, will be approaching infinite.

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

Not everyone. I thought MAGA wanted smaller government. All the offices you guys don't like will now be run by people who don't know what they're doing. You and I will be paying the salaries of thousands of federal employees who will be working hard to destroy those agencies. Their only qualification is loyalty to POTUS/Emperor. Big fan of the Russian invasion? Most people don't like Putin the way you seem to. Do you? It doesn't seem that way. I think you just voted to raise your own taxes to pay for a government loyal to one man. Elephant in the room? That's one of the first indicators for me for anybody's job performance. Most of the people who worked for him the first time refuse to be loyal to him now. They haven't exactly been quiet about how dangerous TFG is. Please elaborate on Kamala Harris' major faults.

Diagrams, Diagrams, Diagrams. Does this sketch help ?

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

y = (((pnp^2/ x ) + x^2) / pnp) pnp = x * y (((((pnp^2 / x) + x^2)) / x) / pnp) where y = 0 If all of these are true in the factors we wish to find, x and y, is there a limit; a range; that could be computed that said if x is this big then y is that big? It wouldn’t be a differential equation that solves a spring. But how do I find and x that is true by testing if y is also true in these 4 constraints? It is a simple idea, but what is the math that completes it? I know that where y on the graph equals zero x has the value approximate to the smaller factor. I have an equation that will tell me y factor knowing x. If you move x larger y gets smaller. Move x smaller y gets larger. There is only a certain range that will prove this true. Combine that with all the other constraints you have and equation that solves a polynomial.

Well she wasn’t chosen Biden had medical problems. Problems that appear to be most of his term. But the things you don’t like in Trump are the same as Harris. I saw a video of Harris where she boasted she could ruin someone’s life with the swipe of a pen. In this video she says that she wouldn’t even have to go to trail, she’d bankrupt them. Same stuff, different person, presented better on evening news.

The "he's a racist" narrative isn't going to work, and there's no litmus test for determining who is a racist and who isn't, beyond common sense which doesn't likely apply to Trump. It's merely a talking point which has been in use since the very beginning, and most likely doesn't relate to actual policy or actions, but rather tangential things. Pointing out that he has racists who support him, for example, is a strategy that has been in use long before Trump ran for office (similar talking points were used against Ron Paul, for example). If you keep up with this, you'll eventually have to try to argue that more than half of Americans are racists, fascists, or Nazis due to having voted for him, and that narrative simply won't fly due to being one of the most common and uneducated talking points used by opposition. It's proven ineffective in two elections, so coming up with something more original is likely in order.