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cotangent

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cotangent
cotangent, n. (a.) Trig. (kəʊˈtændʒənt) [f. co- prefix 4 + tangent. The L. cotangens is used by Gunther Canon Triangulorum, 1620.] The tangent of the complement of a given angle. (Abbrev. cot.)1635 I. W. Sciographia 47 So is the tangent of R. Z. P. To the cotangent of R. P. Z. 1704 Harris (cited by ... Oxford English Dictionary
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Cotangent sheaf
differentials on affine charts to get the globally-defined cotangent sheaf.) Cotangent stack For this notion, see § 1 of A. Beilinson and V. wikipedia.org
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cotangent
cotangent/kəuˈtændʒənt; ko`tændʒənt/ n(abbr 缩写 cot) (mathematics 数) tangent of the complement of a given angle 余切. 牛津英汉双解词典
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Cotangent matrix - Grasshopper - McNeel Forum
16 hours ago — Could you please give me some guidance? I saw it on wikipedia (the L in the pictured formula):. en.wikipedia.org · Discrete Laplace operator. In ...
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Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point Thus it defines a vector bundle on M: the cotangent bundle. Smooth sections of the cotangent bundle are called (differential) one-forms. wikipedia.org
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Cotangent complex
In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric The definition of the cotangent complex The correct definition of the cotangent complex begins in the homotopical setting. wikipedia.org
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Cotangent bundle This may be a poorly phrased question - please let me know of it - but what is the correct way to think of the cotangent bundle? It seems odd to think of it as the dual of the tangent bundle (I am fin...
Your manifold is the configuration space for some system of particles, and the cotangent bundle is then the phases, so the cotangent directions are velocities
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Cotangent bundle tensor product tangent bundle What is the meaning of Cotangent bundle tensor product tangent bundle: $T^*M\otimes TM$? what will an element of this space be?
This is isomorphic to $Hom(TM,TM)$. Namely it is the endomorphism bundle of the tangent bundle. In general given two vector spaces(or bundles) we have an isomorphism between $W \otimes V^*$ and $Hom(V,W)$.
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Trivial Tangent and Cotangent Bundles If we have a smooth manifold $M$, why is the tangent bundle $TM$ trivial (as a vector bundle) iff the cotangent bundle $T^*M$ is trivial as well?
Hint: Given a global frame for either, there is a pretty natural way to create a global frame for the other by dualizing or taking the "covector".
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Cotangent bundle
Smooth sections of the cotangent bundle are differential one-forms. Definition of the cotangent sheaf. Let M be a smooth manifold and let M×M be the Cartesian product of M with itself. The diagonal mapping Δ sends a point p in M to the point (p,p) of M×M.The image of Δ is called the diagonal.
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Cotangent Fields: Exactness vs. Conservation Given a smooth manifold. Then a cotangent field is exact iff conservative: $$\alpha\in\mathcal{X}^*(M):\quad\alpha=\mathrm{d}h\iff\oint\alpha=0$$ How to prove this properly?
_This proof is taken from Lee's Smooth Manifolds._ Consider up to a constant: $$h(x)=\int_z^x\alpha$$ Regard a smooth coordinate path: $$\gamma:[0,\varepsilon)\to M:\quad\hat{\gamma}(t):=(0,\ldots,0,t,0,\ldots,0)\quad(\gamma(0)=z)$$ So one obtains as partial derivatives: $$\frac{\partial h}{\partial...
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Can $S^4$ be the cotangent bundle of a manifold? I am asking the question because in the classical mechanics book by Arnold, he states that there is a distinguished $1$-form on $T^*V$. It seems that there is no such d...
In particular, neither $S^2$ nor $S^4$ can be a cotangent bundle.
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Can $S^4$ be the cotangent bundle of a manifold? I am asking the question because in the classical mechanics book by Arnold, he states that there is a **distinguished** 1-form on $T^*V $. It seems that there is no suc...
No, the total space of a cotangent bundle $T^*M$ is always non-compact, while a sphere $S^{2n}$ is compact.
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System of parameters which have linear independent images in the cotangent space > Given a Noetherian, local ring $(R,m)$, can we always find a system of parameters whose images in the cotangent space $m/m^2$ are line...
By induction let's just worry about picking one parameter element $x$. We need $x$ to be in $m$, but outside all the minimal primes of $R$ and $m^2$. This smells exactly like Prime Avoidance (note that they don't have to be all primes!). Now replace $R$ by $R/(x)$ and repeat.
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Complex structure on cotangent bundle If $M$ is a complex manifold with complex structure $J$, why does the cotangent bundle of $M$ carry a natural complex structure, and not an almost complex structure. Is that obvious?
As Olivier suggests, you can define a complex structure on $T^*M$ directly using the complex structure on $M$. This is basically the complex version of Proposition $3.18$ of Lee's _Introduction to Smooth Manifolds_ (second edition) except that instead of $T^*M$, Lee is using $TM$. If you would like ...
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