Questions tagged [derivations]
42 questions
17
votes
2 answers
A non-trivial derivation on $C^{k}(\mathbb{R})$ for $k\geqslant 1$?
Recall that a derivation on a commutative algebra $A$ is a linear operator $D:A\to A$ which satisfies the Leibniz rule for products $D(fg)=fDg+gDf$ . The standard differentiation $f\mapsto f'$ is certainly a derivation on $C^{\infty }(\mathbb{R})$…
Udo Zerwas
- 599
11
votes
3 answers
Relation between Jacobi identity, commutator, and derivation
Here are some standard facts about these concept: The commutator satisifies the Jacobi identity. The commutator of two derivations is a derivation. The Jacobi identity is equivalent to $\mathrm{ad}_x$ is a derivation (where $\mathrm{ad}_x(y)= [x,y]$…
duck
- 810
6
votes
1 answer
Soft Question - Generalizations of the Derivative
This is a soft question. I'm asking for any interesting and rather unknown generlizations of the derivative. I know it is generalized through derivations which are functions $\delta$ satisfying $$\delta(uv) = v\delta(u) + u\delta(v)$$ Some of them…
Mako
- 1,035
6
votes
0 answers
Is there a combinatorial interpretation of the arithmetic derivative?
The arithmetic derivative is a derivation on $\mathbb{Z}$ that is $1$ for all prime numbers. On positive integers other than 1, this always returns a positive integer. My question is, is there a way to interpret the arithmetic derivative as counting…
Darryl Johnson
- 125
5
votes
1 answer
Existence of $\mathbb{N}$-grading compatible with LNDs.
Let $B$ be a finitely generated integral $\mathbb{C}$ -domain. Let $\partial:B\to B$ be a LND, locally nilpotent derivation , i.e. a $\mathbb{C}$ -linear map satisfying Leibniz rule: $\partial(fg)=f\partial(g)+\partial(f)g$ for $f,g\in B$ ; locally…
Yikun Qiao
- 1,549
4
votes
1 answer
On a Universal Property for the Tangent Space.
A time ago, I was asking if there exists an universal property of the tangent space and what it says about any construction of it. I've found the definition maded in Tammo Dieck's book of Algebraic Topology: A tangent space at $p$ is a vector space…
Pauli
- 1,217
3
votes
0 answers
Tangent space and derivation in infinitely dimensional manifolds
For an $n$ -dim differentiable manifold, the tangent space is identified with derivations. Is this still true for infinitely dimensional manifolds? On $\mathbb{R}^n$ , the proof for the proposition that map $\tau:D_v\to \partial_v$ from directional…
user760
- 3,088
3
votes
0 answers
Fuzzy pictures of schemes, nilpotents, and associated primes
In Exercise 6.6.U (a) of Ravi Vakil's FOAG, we are given a scheme $X = \mathbb{C}[x, y]/I$ for some unknown $I$ . The one thing we're told about $I$ is the associated points. The exercise then asks us to draw a picture of $X$ , including "fuzz". In…
3
votes
1 answer
Are there non-trivial $\mathbb{Z}$-linear derivations over the real numbers?
Given the algebraic definition of a $\mathbb{Z}$ -linear derivation over a commutative ring $D:R\to R$ with $D(a+b)=D(a)+D(b)$ and $D(a \cdot b)=D(a) \cdot b+a \cdot D(b),$ there is always the trivial derivation where $\forall_{a\in R} \,D(a)=0$ . I…
Gyro Gearloose
- 1,416
3
votes
0 answers
Tangent sheaf and Tangent space
A few days ago, I came across the notion of the tangent sheaf of an affine $k$ -scheme $X = \operatorname{Spec} A $ , with $k$ an arbitrary field. The natural question that arose was: what is the relationship between this and the tangent space…
3
votes
2 answers
Confusion on Notations of Partial Derivatives on Manifolds
I'm confused with the notations of partial derivative on manifolds in Tu's An Introduction to Manifolds .. Just to make clear the notations I'm using, what I've known and which part I'm confusing, please let me to restate the question formally. Let…
TheHan6edMan
- 31
3
votes
2 answers
How to derive an answer from Implicit Differentiation to another answer?
When we find $dy \over dx$ of the equation ${1 \over x} + {1\over y} = x - y$ , we can differentiate both sides to obtain: ${dy \over dx} = {y^2(x^2 + 1)\over x^2(y^2-1)}$ ...(1) On the other hand, we can first transform the equation into $y + x =…
user528789
2
votes
0 answers
Given a graded vector space and a graded linear operator on it, can we always define a product on the vector space that makes it a derivation?
Given a graded vector space $V = \bigoplus_{i=0}^{\infty} V_i$ and a graded linear operator $D$ on it where $D(V_0) = \{0\}$ and $D(V_{i+1}) \subset D(V_{i-1})$ , is it always possible to define a product on $V$ that turns it into a graded algebra…
Dale
- 501
2
votes
1 answer
How was this $y\left(y’’+\frac{1-p}vy’\right)-(y+1-p)y’^2=0$ power series recurrence derived?
In György Steinbrecher’s and William Shaw’s Quantile mechanics $(47)$ to $(51)$ , it can be found that: $$\begin{aligned}y\left(y’’+\frac{1-p}vy’\right)-(y+1-p)y’^2=0,y(0)=0,y (0)=1\iff (1-p)y’^2=\frac{1-p}vyy’+yy’’-yy’^2\\\implies…
Тyma Gaidash
- 14,281
2
votes
0 answers
An adjunction involving connections by analogy with derivations
I am trying to figure out whether the concepts of connection and flat connection can be defined in a way analogous to how derivations are defined here by Akhil Mathew. As is discussed at the link, derivations can be understood using free abelian…
user900250