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To derive the Bernoulli equation for inviscid fluids, the plan is to rewrite the Euler equation in such a way that we have gradients. I'll write the Euler equation with gravity here $$\frac{\partial \vec{u}}{\partial t} + \vec{u} \cdot \vec{\nabla} \vec{u} = -\frac{1}{\rho} \vec{\nabla} p + \vec{g}.$$ Recall $g = - \vec{\nabla} \Psi$, and $\vec{u} \cdot ... 2 1) The relation$\frac{dF}{dS}=d\left(\frac{F}{S}\right)$is certainly incorrect as @Floris has mentioned in the comment. As the simplest counter-example, consider a linear function,$F(S) = \alpha \, S$, with$\alpha \neq 0$as a proportionality constant. Then, one could easily see that$$\frac{dF}{dS}= \alpha \neq 0 = d\left(\frac{F}{S}\right) = ... 1 Suppose you have two masses M1(=M) and M2(=M) with volumes V1 and V2, respectively. Then the total density is the total mass divided by the total volume. So$\rho_{mix}$=2M/(V1+V2). V1=M/$\rho_1$and V2=M/$\rho_2$so$\rho_{mix}$=2M/(M/$\rho_1$+M/$\rho_2\$), which after canceling the M's and simplifying the expression is equal to what you wrote above. ...

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In the usual model of black hole, with general relativity only and no quantum mechanics, yes, once passed a threshold (the center of) a star collapse down to an infinitely dense singularity, because there are no longer any repulsing force to counterbalance auto-gravity. But then if you add quantum mechanics then such exact concentrated locality is not ...

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Knowing a specific density is rather handy, because it allows us to relate the volume and the mass of a substance. But to calculate the pressure, the density isn't necessary. What is needed is the mass. If the fluids mix, the volume (and therefore the density) may change, but the mass does not. The total mass remains constant. If prior to mixing, the ...

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Because conventionally we assume constant temperature, and length and density are also assumed to be constants for a given resistor. Of course, this is not true. In some circuit designs I have to pay very careful attention to resistance changes with temperature, and indeed this is sometimes used to provide temperature measurements in the form of RTD ...

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