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Questions tagged [differentiation]

Differentiation is the set of techniques and results from Differential Calculus, concerning the calculation of derivatives of functions or distributions.

-1 votes
1 answer
55 views

What happens if we differentiate spacetime with respect to time? [closed]

Essentially, what would differentiating space-time with respect to time provide us with? What are the constraints associated with such operations? Is it possible to obtain a useful physical quantity ...
Kimaya Deshpande's user avatar
0 votes
0 answers
35 views

Changing coordinate system [migrated]

Someone please explain how did we get second term in equation 2.15.
Mr. Wayne's user avatar
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0 votes
0 answers
41 views

What are the operators here and how are these formulas derived? [closed]

In (23), are grad and div some kind of scalar operators comparing to $\nabla$ and $\nabla\times$? because tbh I dont know how $\text{curl}(\mu^{-1}\text{curl}\textbf{A})$ turns into $\text{div}\mu^{-1}...
user900476's user avatar
0 votes
1 answer
140 views

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

What's the difference? $$\nabla_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~\text{ and }~\partial_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~?$$ In John Dirk Walecka's book 'Introduction to General Relativity',...
Jianbingshao's user avatar
6 votes
3 answers
1k views

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

Here are the equations. ($V$ represents a potential function and $p$ represents momentum.) $$V(q_1,q_2) = V(aq_1 - bq_2)$$ $$\dot{p}_1 = -aV'(aq_1 - bq_2)$$ $$\dot{p}_2 = +bV'(aq_1 - bq_2)$$ Should ...
Bradley Peacock's user avatar
0 votes
1 answer
75 views

Confusion about contraction and covariant derivatives [closed]

Understanding Contraction and Second Covariant Derivatives in Tensors I am confused about contraction in tensors and the second covariant derivative in tensors. Consider a tensor $T_{\mu\nu}$ and the ...
Yuv Agarwal's user avatar
1 vote
3 answers
85 views

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

I'm working through Chap. $30$ of Dirac's "GTR" where he develops the "comprehensive action principle". He makes a very slick and mathematically elegant argument to show that the ...
Khun Chang's user avatar
2 votes
1 answer
59 views

Total differential of internal energy $U$ in terms of $p$ and $T$ using first law of thermodynamics in Fermi's Thermodynamics

While reading pages 19-20 of Enrico Fermi's classic introductory text on Thermodynamics, I ran into two sources of confusion with his application of the First Law. Fermi introduces a peculiar notation ...
user104761's user avatar
1 vote
1 answer
61 views

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

I want to experiment with this relation (from Dirac's "General Theory of Relativity"): $$T^{μν} = -\left(2 \frac{∂L}{∂g_{μν}} + g^{μν} L \right)$$ using the electromagnetic Lagrangian $L = -(...
Khun Chang's user avatar
0 votes
1 answer
46 views

Transformation to replace a Material derivative with a spatial derivative

In the technical paper referenced below, Gringarten et al. claim that the transient energy transport equation in a planar conduit (Eq. 1 in their paper) $$ \rho c \Bigg[ \frac{\partial T(z,t)}{\...
Sharat V Chandrasekhar's user avatar
1 vote
3 answers
102 views

The conservative force [closed]

I read about the definition of the curl. It's the measure of the rotation of the vector field around a specific point I understand this, but I would like to know what does the "curl of the ...
Dirac-04's user avatar
2 votes
0 answers
45 views

Covariant (absolute) derivative of a vector along a curve -- compare cartesian vs. polar coordinates [migrated]

BACKGROUND: Suppose $A^μ$ is a vector field and $x^μ(λ)$ is a curve in spacetime. A first guess at measuring the change in $A^μ$ along the curve might be $$\frac{dA^μ(x(λ))}{dλ} = \frac{∂A^μ}{∂x^ν} \...
Khun Chang's user avatar
0 votes
1 answer
85 views

Differential form of Lorentz equations

A Lorentz transformation for a boost in the $x$ direction ($S'$ moves in $+x$, $v>0$) is given by: $$ t'=\gamma\left(t-v\frac{x}{c^2}\right),~x'=\gamma(x-vt)$$ In the derivation of the addition of ...
ceciled's user avatar
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