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2

Presumably you're okay with the intermediate step $$ \int 2\vec h\cdot \vec r_\text{cm} dm = 2\int \vec h\cdot \vec r_\text{cm} dm \tag1 $$ since the number two is a constant, and integration is linear over a constant. The authors assert that $\vec h$ is also constant, so it comes out as well. If it's motion of the dot product across the integral sign ...


0

The derivation is not correct. The mass within the control volume is $\rho A\Delta x$. The rate of energy accumulation within the control volume is $\rho A\Delta xC\frac{\partial T}{\partial t}$. So the heat balance should be:$$\rho A\Delta xC\frac{\partial T}{\partial t}=Q_x-Q_{x+\Delta x}$$Dividing by $\Delta x$ and taking the limit as $\Delta x$ ...


1

For a force to do work, it must act upon an object as that object moves some distance (with a component along or opposed to the direction of that force). Holding an object stationary, but carrying the strength of a given force, adds no more work done. Because of this, we say that $$W=\int_C\vec{F}\cdot d\vec{r}$$ where C is the curve in space along which ...


1

This is because $P(t)$ as stated is the instantaneous power as a function of time and $W =\mathbf F\Delta \mathbf x$ holds only for constant forces. More generally, recall that a definition of work is the integral: $$W = \int_C\mathbf F(x)\mathrm d\mathbf x$$ Where $C = C(x,t)$ is some curve in space/time. Expressing in terms of time gives: $$W = ...


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The (Lie-)group $U(1)$ is the topological space $S^1$ (what we call a circle together with its standard open subsets) together with a rule how to multiply its points. In its representation as numbers in ${\mathbb C}$ with absolute value $1$, we have ${\mathrm e}^{{\mathrm i}\alpha}\bullet{\mathrm e}^{{\mathrm i}\beta}:={\mathrm e}^{{\mathrm ...



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