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16 votes

In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?

The notation is a little sloppy from a purely mathematical point of view (although common in physics) so it might be causing a little confusion. To help clarify, it might help to use different letters ...
Andrew's user avatar
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3 votes

In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?

They are partial derivatives. From the chain rule, we have $$ \frac{\partial V(aq_1-bq_2)}{\partial q_1}= a V'(aq_1-bq_2),\\ \frac{\partial V(aq_1-bq_2)}{\partial q_2}= -b V'(aq_1-bq_2). $$ For ...
mike stone's user avatar
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2 votes
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Total differential of internal energy $U$ in terms of $p$ and $T$ using first law of thermodynamics in Fermi's Thermodynamics

Fermi was probably the last physicist who excelled as a theoretician and an experimentalist. His lectures inspired many students, who became outstanding physicists in turn. His textbook on ...
GiorgioP-DoomsdayClockIsAt-90's user avatar
2 votes

How do you differentiate $F^{αβ}$ with respect to $g_{μν}$?

If you are going to differentiate $L$ with the respect to the metric, $L$ needs to be rewritten without the metric being implicitly used anywhere. Otherwise, you will not vary the entire dependence on ...
William Elderfield's user avatar
2 votes
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Transformation to replace a Material derivative with a spatial derivative

In the technical paper referenced below, Gringarten et al. claim... $$ \rho c \Bigg[ \frac{\partial T(z,t)}{\partial t}+v\frac{\partial T(z,t)}{\partial z}\Bigg]=Q(t) $$ can be reduced with the ...
hft's user avatar
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2 votes

In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?

No. The notation means that the V on the RHS is a function only of one variable, and so its derivative is the simplest, one-variable derivative.
naturallyInconsistent's user avatar
2 votes
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How to expand $(D_\mu\Phi)^\dagger(D^\mu\Phi)$ in $SU(2)$?

You appear to not appreciate your expression as a row vector dotted on a column vector (possibly sandwiching operators). I corrected your expression to $$\partial_\mu\Phi^\dagger \partial^\mu \Phi - \...
Cosmas Zachos's user avatar
2 votes

What happens if we differentiate spacetime with respect to time?

I think what you're reaching for is the four vector velocity of an object. That is, differentiating it's position with respect to time. You started with a four dimensional vector (position) and ...
StephenG - Help Ukraine's user avatar
1 vote

Why is $(\partial_\mu F_{\alpha\beta})F^{\alpha\beta}=F_{\alpha\beta}\partial_\mu(F^{\alpha\beta})$?

Having realized my error of thinking the multiplication was non-commutative, it becomes clear: $$(\partial_\mu F_{\alpha\beta})F^{\alpha\beta}=F^{\alpha\beta}(\partial_\mu F_{\alpha\beta})=\eta_{\...
user410662's user avatar
1 vote
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Why is $(\partial_\mu F_{\alpha\beta})F^{\alpha\beta}=F_{\alpha\beta}\partial_\mu(F^{\alpha\beta})$?

Here the solution asserts that this is equal to simply equal to twice the first part of the term, implying $(\partial_\mu F_{\alpha\beta})F^{\alpha\beta}=F_{\alpha\beta}\partial_\mu(F^{\alpha\beta})$. ...
hft's user avatar
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1 vote

What's the difference? $\nabla_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~\text{ and }~\partial_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~?$

The expression $\partial_\mu \mathbf{e}_\nu$, at face value, does not really make sense in curved space due to the vectors being in different tangent spaces. In order to make it work, vectors need to ...
Vincent Thacker's user avatar
1 vote

Does (covariant) divergence-freeness of the stress-energy tensor ${T^{\mu\nu}}_{;\nu}=0$ follow from the Bianchi identity?

The law of local conservation of $T$ is consequence of the law of motion of the matter part of the total action matter + curvature $$I[g]+ J[\phi, g]$$ These equations of motion for the matter arise ...
Valter Moretti's user avatar
1 vote

The conservative force

While @basics has the math right, I understand your question is about the physical interptetation. To answer this, we must understand what a force field is, since the definition of the rotation ...
paulina's user avatar
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1 vote

The conservative force

An example might be the best way to see it. Set up some electrodes that create a horizontal electric field. Make it so the field strength is proportional to height. This is a non-conservative field. ...
mmesser314's user avatar
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1 vote

The conservative force

Remark the curl it's the measure of the rotation of the vector field around a specific point Before answering your question, I'd replace (local) "rotation" with (local, or elementary) &...
basics's user avatar
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