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Products are omnipresent in physics – and even in less quantitative sciences – from Day One. The product $A=XY$ of two numbers may be visualized as the area $A$ of the rectangle whose sides are the two factors $X$ and $Y$. When embedded in physics, this simple formula $A=XY$ already includes units: the sides are in meters and the area is in squared meters. ...

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If we take your same two masses and write $m_1 = s \cdot m_2$, it's easy to understand what the scalar multiplication is doing: it's scaling. $m_1$ is $s$-times as big as $m_2$. In your language, multiplying $m_2$ by $s$ gives us a new mass which has the mass of $s$ [many] $m_2$'s. This may not seem to answer your question about work, because now the things ...

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It's a case of bad labeling: the $i$,$j$ labels in Fig.1 and Eqs.(4-5) have different meaning. In addition, subscript 1 was dropped on all $B$'s in Eq.(5). Other than that, it's straightforward algebra: Start by rewriting the final result of Fig.(1) in the familiar operator-product form, expand, and rearrange:  \overline{\left[ E \cos(B_1\tau) - i {\hat ...

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Apologies for not producing a most general answer for arbitrary Lie groups, (which you might tease with great effort out of WP ), but only a trail-map for your particular (charmed!) problem. I call it charmed because it should remind you of the Lorentz group, with a,b,c parameterizing Kx,Ky,Kz boosts and d,e,f the three J rotation angles. Decent treatments ...

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On dot product you get magnitude, in units of product of operands. On cross product you get vectors with direction , in units of product of operands.

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'Mathematical Physics' by Kusse and Westwig is just the thing you need. The fifth chapter is devoted to the Dirac-delta function. The book is fairly easy to understand and provides the proofs of the theorems that are stated in Arfken-Weber. After having read this, you can read the appendices I and II in Cohen-Tannoudji (Quantum Mechanics) on Fourier ...

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The first thing that must be said is that the question is not really specific enough: Applications to what exactly are you looking for? To me, a book on algebraic geometry and mirror symmetry, and how it relates to mirror symmetry as physicists know it, is very relevant and interesting. However, I have the feeling that this is not exactly what you're looking ...

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