# Questions tagged [differential-operators]

In mathematics, a differential operator is an operator defined as a function of the differentiation operator.

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### laplacian of a 2d tensor which only has one independent component

Could anyone guide me on how to find the following laplacian in polar coordinate? I have spent an hour searching for a reference but could not find anything helpful. T is a $2\times2$ tensor in polar ...
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### Why does the d/dx operator output a derivative, whereas the integral operator accepts a differential?

Say I have differentiation operators $D_1$ and $D_2$, where $D_1$ is the normal gradient operator, and $D_2$ yields a differential $$D_2 \Big{[} y = sin(x) \Big{]}$$ $$dy = cos(x) dx$$ It would seem ...
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### Proof of invariance of propagators or Green functions from their defining property rather than an explicit expression (Klein-Gordon eq.)

This same question has been asked in the physics section and answered with the usual physics proof, which is unsatisfying to me as I wish to derive the symmetry properties DIRECTLY FROM THOSE OF THE ...
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### Does an almost complex structure $J$ induce a differential operator on $TM$?

Let $M$ be an almost complex manifold with almost complex structure $J$. We have that $J : TM \to TM$ is a vector bundle isomorphism. Naturally, this induces a map on sections, and with abuse of ...
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### Interpreting Taylor series as exponentiating an operator

In this video at 46:11, the Taylor series of function is rewritten a very different way by using exponential operator. I'll describe the method used:  f(x) = \sum_{i} \frac{(x-a)^i}{i!} \frac{d^i f(...
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### Transformation law of elliptic operators

I am trying to understand elliptic operators and how they transform on Riemannian manifolds. It is stated in Yoshida's Functional Analysis (page 425) that if the local coordinate expression of a ...
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