Questions tagged [differentials]

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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 ...
Qfwfq's user avatar
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25 votes
2 answers
1k 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 ...
Toby Bartels's user avatar
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21 votes
1 answer
2k 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 $df=0$,...
Allen Knutson's user avatar
9 votes
2 answers
2k 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. ...
QcH's user avatar
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9 votes
3 answers
1k views

If $\Omega_{X/Y}$ is locally free of rank $\mathrm{dim}\left(X\right)-\mathrm{dim}\left(Y\right)$, is $X\rightarrow Y$ smooth?

Suppose I have a morphism $f:X\rightarrow Y$ such that the relative sheaf of differentials $\Omega_{X/Y}$ is locally free. Does it follow that $f$ is smooth? The answer is no, but for a silly reason. ...
Anton Geraschenko's user avatar
6 votes
1 answer
644 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 ...
H. Arponen's user avatar
5 votes
2 answers
1k 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 ...
Will Chen's user avatar
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4 votes
3 answers
512 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 ...
user2330's user avatar
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3 votes
2 answers
892 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?
Peter Lee 's user avatar
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3 votes
0 answers
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Logarithmic differentials on an arithmetic surface, and Poincaré residue

Suppose that $X$ is an arithmetic surface, i.e. $\pi: X \to S$ flat and relative dimension 1 over a Dedekind scheme $S$, and assume $X$ smooth. Let $Y \subset X$ be a horizontal effective Cartier ...
PrimeRibeyeDeal's user avatar
3 votes
0 answers
118 views

Is the cohomology $H^1(X, \mathcal{E}^\nabla)$ trivial, for the sheaf of constants of an algebraic connection $\nabla$?

Suppose that $\pi:X\to S$ is a flat morphism between Noetherian, integral schemes (of characteristic zero, if need be). Let $\mathcal{E}$ be a locally free sheaf on $X$, and $$\nabla:\mathcal{E} \to \...
PrimeRibeyeDeal's user avatar
3 votes
0 answers
2k views

Derivative of the regularized upper incomplete gamma function

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} \Gamma(x)&=&\int_0^\...
ppyang's user avatar
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2 votes
3 answers
2k views

Optimum small number for numerical differentiation

The Wikipedia article on numerical differentiation mentions the formula $$ h=\sqrt \epsilon \times x $$ where $\epsilon$ is the machine epsilon (approx. $2.2\times 10^{-16}$ for 64-bit IEEE 754 ...
Joonas Pulakka's user avatar
2 votes
2 answers
579 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 $\...
Don Stanley's user avatar
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2 votes
1 answer
410 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 ...
Martin Brandenburg's user avatar
2 votes
2 answers
922 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. ...
Doogies's user avatar
  • 21
2 votes
1 answer
506 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 $...
Zev Chonoles's user avatar
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1 vote
1 answer
286 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 \...
H. Arponen's user avatar
1 vote
1 answer
220 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 $\...
dhagbert's user avatar
  • 671
1 vote
1 answer
538 views

Choosing Notation for Variable Substitution into Derivative Expressed with Differentials [closed]

Consider function $f(x)$. I've counted 4 possible notations to write a derivative of $f(x)$ at point $x = a$: $f'(a)$; $\frac{\operatorname{d}{f(a)}}{\operatorname{d}x}$; $\left.\frac{\operatorname{d}...
Alexander Shukaev's user avatar
1 vote
1 answer
204 views

Third order matrix differential norm

Suppose we have a function $f:\mathbb{R}^n\to\mathbb{R}$ that is at least three times differentiable. Clearly, there is a relationship between the symmetric trilinear form $$T_1=\nabla^3f(x),$$ and ...
RS-Coop's user avatar
  • 39
0 votes
1 answer
275 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 ...
user17414's user avatar
0 votes
1 answer
265 views

determinant of integrals of forms

Let $A$ be a complex abelian variety of dimension $d$. Let $\omega_1, \ldots, \omega_j \in H^0(A, \Omega^1_A)$ be linearly independent (so $j \leq d$) and consider $\gamma_1, \ldots, \gamma_j \in H_1(...
detted92's user avatar
0 votes
0 answers
137 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 $M$ ...
Qfwfq's user avatar
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0 votes
0 answers
584 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 $...
Will Chen's user avatar
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