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1answer
144 views

Contraction of graded vector fields on de Rham complex

Given a commutative algebra $A$ smooth over a field $k$ of characteristic zero, the module of K\"ahler differentials $\Omega^{1}$ is projective of finite rank and so the sum of all wedge powers ...
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0answers
432 views

Derivative of the regularized upper incomplete gamma function

Hello everyone! I have a question about the derivative of the regularized upper incomplete gamma function. Considering the gamma function and the incomplete gamma function \begin{eqnarray} ...
6
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1answer
500 views

ANOTHER Exterior differential system on $SO(3;\mathbb R) \times \mathbb R$

I have another exterior differential system for one forms $U^i$, where the $\theta^i$ are a cotangent basis on $SO(3)$, i.e. they satisfy $d \theta^i = \epsilon_{ijk} \theta^j \wedge \theta^k$ for the ...
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1answer
171 views

Exterior differential system on $SO(3;\mathbb R) \times \mathbb R$

I have the following exterior differential system for one forms $\alpha, \beta, \gamma$, where the $\theta^i$ are a cotangent basis on $SO(3)$, i.e. they satisfy $d \theta^i = \epsilon_{ijk} \theta^j ...
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0answers
113 views

Closed forms and trajectories of vector fields

This question is inspired by this recent one and this one; I hope it's not too elementary. Let $M$ be a (closed) smooth manifold and $X$ a vector field on $M$. Fix any Riemannian metric $g$ on $M$ ...
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0answers
222 views

what does it mean for a differential to be regular at a singular point?

Let $\omega$ be a differential form on a singular integral curve $X'$ over some algebraically closed field $k$ (ie, $\omega$ is an element of the stalk of the sheaf of differentials $\Omega_{X'}$ of ...
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2answers
369 views

trying to understand the support of the sheaf of relative differentials

So I'm trying to understand a proof of Belyi's theorem from http://eprints.soton.ac.uk/29785/1/b45h1koe.pdf specifically lemma 3.4. The setup is as follows: Let $X/\mathbb{C}$ be a curve, and let $t ...
5
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2answers
552 views

Relationship between double tangent bundle, exterior derivative and connection

I am totally new to the subject differential geometry, and that probably reflects itself in the naive question that I'm trying to formulate. I hope this question does not get closed because of this. ...
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2answers
488 views

Geodesics for a Cone Metric

Here is a question that I hope/suspect is elementary but cannot find a reference for. Suppose we are given a surface, S, with a conformally Euclidean metric, |f(z)||dz|, where f(z) is meromorphic. ...
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0answers
206 views

Partial differentiability and Continuity [closed]

If a function has, say, partial derivatives up to order $n$, can you conclude continuity of some or all derivatives of lower order? Especially, if a function has partial derivatives of any order is ...
16
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1answer
726 views

When does the relative differential $df=0$ imply that $f$ comes from the base?

Let $A \to B$ be a map of commutative rings, and $d : B \to I/I^2$ be defined by $df = f\otimes 1 - 1\otimes f$, where $I$ is the kernel of $B \otimes_A B \to B$, as in [Hartshorne II.8]. If ...
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1answer
199 views

Compute differential on cotangent bundle

Hi, This is my question. Can we compute easily the differential of the following map ? $$ f:(x,\xi^\star)\in TS^{2n-1} \mapsto \xi^\star(ix)\in \mathbb{R} $$ where $TS^{2n-1}$ is ...
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1answer
1k views

Various flavours of infinitesimals

I'm not sure if this is a soft question, and whether it may be too broad or, on the contrary, too localized. Well, in Mathematics the concept of "infinitesimal" has been of extreme importance for ...
2
votes
1answer
310 views

Kahler differentials of a hypersurface over a non-algebraically closed field

The following was recently on my algebraic geometry homework: Let $k$ be an algebraically closed field, $f\in B=k[x_1,\ldots,x_n]$, and $A=B/(f)$. Show that $\Omega_{A/k}$ is locally free of rank ...
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2answers
789 views

Is there a convenient differential calculus for cojets?

I understand the basics of exterior differential geometry and how to do calculus with exterior differential forms. I know how to use this to justify the notation dy/dx as a literal ratio of the ...
2
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3answers
920 views

Optimum small number for numerical differentiation

http://en.wikipedia.org/wiki/Numerical_differentiation#Practical_considerations mentions the formula $h=\sqrt \epsilon * x$ where $\epsilon$ is the machine epsilon (approx. 2.2e-16 for 64-bit IEEE ...
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3answers
422 views

Is there a reason for defining the differential forms before the vector fields ?

Hi, my question is the following : In EGA IV chapter 16, given $X$ a scheme over $S$, Grothendieck defines first $\Omega^1_{X/S}$, the $O_{X}$-module of the 1-differentials. He then defines the ...
2
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1answer
352 views

derivative in the ring k[e]/e², chain rule

Let $k$ be a ring and $\overline{k} = k[\epsilon]/\epsilon^2$. For every $f \in k[t]$ there is a unique $f' \in k[t]$ such that $f(t+\epsilon)=f(t)+\epsilon f'(t)$ holds in $\overline{k}[t]$. It ...
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2answers
438 views

Resolution of a free lie algebra as a module over its universal enveloping algebra.

Let $L=L(V)$ be a free Lie algebra on a vector space $V$ and $A=T(V)$ the tensor algebra. Make $L$ into a module over $A$ consistent with the formula $a\cdot \alpha=[a,\alpha]$ for $a\in V$ and ...
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2answers
680 views

How to introduce Kahler differential in category? [closed]

How to define Kahler differential in an abelian category or more general category? Say exact category? Is there any interesting example?
5
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3answers
474 views

If \Omega_X/Y is locally free of rank dim(X)-dim(Y), is X->Y smooth?

Suppose I have a morphism f:X→Y such that the relative sheaf of differentials ΩX/Y is locally free. Does it follow that f is smooth? The answer is no, but for a silly reason. You could ...