Questions tagged [derivations]
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43 questions
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Understanding the application of FTC in the proof that $\dim \left( T_{p}M \right) = n$
Consider the following proof I am trying to break down
There are two things I don't understand in the proof.
I do not understand the way the author uses the Fundamental Theorem of Calculus. I know ...
- 205
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Confusion about differentiating with respect to local coordinates, and dimension of $T_{p}M$
Background and definitions:
At the moment I am taking a basic course discussing Lie groups and Lie algebras. In the last lecture we have defined the following;
Let $R$ be ring and let $A$ be an ...
- 205
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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 ...
- 501
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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 ...
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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 $\mathrm{ad}_x$...
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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 ...
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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 ...
- 1,406
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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 ...
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$\exp(\mathrm{Der}(L)) \subseteq \mathrm{Aut}(L)$ implies $\mathrm{Der}(L) \subseteq \mathrm{Lie}(\mathrm{Aut}(L))$
Let $L$ be a (finite-dimensional real) Lie algebra. Let $\mathrm{Aut}(L) \subseteq \mathrm{GL}(L)$ be the set of all Lie algebra automorphisms on $L$, and let $\mathrm{Der}(L) \subseteq \mathfrak{gl}(...
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Why is Lie algebra $[L_{\lambda}, L_{\mu}]$ contained in $L_{\lambda+\mu}$
This is for the 2nd part of question 9.8 in the book "Introduction to Lie Algebras" by Karin Erdmann and Mark J. Wildon.
A generalized eigenspace of derivation $\delta$ is defined as as $L_\...
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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 ...
- 599
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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, ...
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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 ...
user528789
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Why don't I see "vector-valued vector fields"?
Let $P\xrightarrow{\pi}M$ be a principal $G$-bundle with connection $\omega\in \Omega^1(P;\mathfrak{g})$.
When I am studying these things, there are Lie-algebra valued differential forms all over. ...
- 1,042
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Derivation of two equivalent functions yields different derivation result. Why?
$f(x)=x+\frac{2x^3}{3-2x}-x^2-1 \\
g(x)=\frac{x(3-2x)+2x^3-x^2(3-2x)-3+2x}{3-2x}$
$f'(x)=\frac{-16x^3 + 46x^2 -30x+9 }{(3-2x)^2}\\
g'(x)=\frac{-16x^3 + 50x^2 -34x+9 }{(3-2x)^2}$
Why $f'(x)$ and $g'(x)$...
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