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Physical motivation for differentiation under the integral

I am thinking about the mathematical process of "differentiating underneath the integral", i.e. applying the theorem $$\partial_s \int_{-\infty}^\infty f(x,s)\,dx=\int_{-\infty}^\infty \partial_s ...
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Is it correct to sum over either index of the metric the same way?

I don't know if the following is correct, i want to compute the following derivative $$\frac{\partial }{\partial (\partial_{\mu}A_{\nu})}\left(\partial^{\alpha}A^{\beta}\partial_{\alpha}A_{\beta} ...
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103 views

Scale-invariant differential operator

For example, the differential operator Laplacian is $$\nabla^2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}.$$ My questions are: Is it scale-invariant? what is ...
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115 views

Implicit Differentiation, A doubt

$v=v_c(\tau, t)$ is a smooth function and suppose we have a relation $y_c(\tau,v_c;t)=0$ when $x_c$ is written in the form $x_c=c+ty_c(\tau,v_c;t)$, $c$ is real constant, $t$ is real number denotes ...
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How to derive the schwarzchild metric?

I'm having trouble differentiating the following when making a change of co-ordinates to determine the Schwarzchild metric. $$r'^{2}=r^{2}C(r)$$ Then taking the total derivative of both sides, the ...
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How to do this index notation differentiation?

I am studying classical Maxwell fields and I am stuck on this differentiating part. How can I derive the result given below ? $$\dfrac{\partial}{\partial(\partial A_{\mu}/\partial x_{\nu})} ...
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45 views

Vector Derivative Transport Theorem Application

I have a position vector in frame A, the derivative of which I want to take relative to an observer in frame B. I apply the Vector Derivative Transport Theorem. The obtained velocity vector is left in ...
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105 views

Why does the cross derivative of the partition function disappear here?

They state that the chemical potential in a canonical ensemble is given by: $$\mu = -kT \frac{\partial{\ln Z(N,V,T)}}{\partial{N}} \tag{1}$$ But if I use the definition of chemical partial (which I ...
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Nicholas Kollerstrom article on the history of Calculus

Today, Newton┬┤s birthday, I read an article posted in the arXiv by Nicholas Kollerstrom http://www.arxiv.org/abs/1212.2666 That basically claims that Newton did not invent Calculus. The article does ...