# What is Derivations: Definition and 106 Discussions

In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field K, a K-derivation is a K-linear map D : A → A that satisfies Leibniz's law:

D
(
a
b
)
=
a
D
(
b
)
+
D
(
a
)
b
.

More generally, if M is an A-bimodule, a K-linear map D : A → M that satisfies the Leibniz law is also called a derivation. The collection of all K-derivations of A to itself is denoted by DerK(A). The collection of K-derivations of A into an A-module M is denoted by DerK(A, M).
Derivations occur in many different contexts in diverse areas of mathematics. The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on Rn. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor algebra of a manifold. It follows that the adjoint representation of a Lie algebra is a derivation on that algebra. The Pincherle derivative is an example of a derivation in abstract algebra. If the algebra A is noncommutative, then the commutator with respect to an element of the algebra A defines a linear endomorphism of A to itself, which is a derivation over K. An algebra A equipped with a distinguished derivation d forms a differential algebra, and is itself a significant object of study in areas such as differential Galois theory.

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1. ### I About derivations of lie algebra

Please, I am looking for a simple example of derivation on ##sl_2##, if possible, I try to use identity map, but not work with me, A derivation of the Lie algebra ##\mathfrak{g}## is a linear map ##\delta: \mathfrak{g} \rightarrow \mathfrak{g}## such that ##\delta([x, y])=[\delta(x), y]+[x...
2. ### I Need to resort to spherical wavefront to derive the LTs?

I have been reading Wikipedia’s derivations of the Lorentz Transformations. Many of them start with the equation of a spherical wavefront and this reasoning: - We are asked to imagine two events: light is emitted at 1 and absorbed somewhere else at 2. For a given reference frame, the distance...
3. ### Quantum Advanced Quantum Mechanics Textbooks: Derivations of Equations

Hi I’m looking for a textbook that shows the derivations of equations such as the different forms of the schrodinger equation fully and step by step.
4. ### Worth learning complex exponential trig derivations in precalc?

This is a pedagogical /time management / bandwidth / tradeoff question, no argument that learning the complex exponential derivation is valuable, but is it a good strategy for preparing for first year Calculus? my 16YO son is taking AP precalc and AP calc next year and doing well, but struggled...
5. ### I Understanding Derivations and Tangent Spaces on Manifolds

Hello! According to the attached proposition on ##C^\infty## manifold space of derivations ##D_m M## is isomorphic to Tangent space ##T_m M##. Cited here another proposition (1.4.5) states the following 1. For constant function ##D_m(f)=0## 2. If ##f\vert_U=g\vert_U## for some neighborhood...
6. ### I Precausality and continuity in 1-postulate derivations of SR

[Moderator's note: Thread spun off from previous one due to topic shift.] Please forgive my ignorance, I've never studied group theory systematically up to now, so I'm not aware of all the concepts and symbols that have been used up to now. Yet, I'm interested in the derivation of the Lorentz...

41. ### MHB What is the Definition of a Derivation for a Lie Algebra?

Loosely speaking a derivation D is defined as a function on an algebra A that has the property D(ab) = (Da)b + a(Db). Now, if we define the map ad_x: y \mapsto [x,y] and apply this to the Jacobi identity we get ad_x[y,z] = [ ad_x(y),z ] + [ y, ad_x(z) ] . This does not look quite like the...
42. ### Lie bracket of derivations in space of r-forms

Hello In textbook by Kobayashi and Nomizu derivation of rank k in space of all differential forms on a manifold is defined to be operator that is linear, Leibnitz and maps r-forms into r+k-forms. By Leinbitz I mean, of course: D(\omega \wedge \eta)=(D \omega) \wedge \eta + \omega \wedge (D...
43. ### Should I memorize all the derivations in QM?

Hi. I'm a self-learner, and I'm doing an introductory QM from Griffiths. My question is should I memorize all the derivations and proofs in the book or I just have to memorize the final results? Another point is that I have nobody to ask about my solutions to problems or to help me in the hard...
44. ### Derivations - What's Acceptable?

Homework Statement Hello, this isn't really a h/w problem that fits the template, but here seems the most sensible place to post. I am writing a paper (not original research) and part of the requirements is that I include a "comprehensive discussion of the relevant theory (including...
45. ### Derivations vs Directional derivatives

In some books, when discussing the relation between partial/directional derivatives and tangent vectors, one makes a generalization called a "derivation". A derivation at ##\vec{a} \in \mathbb{R}^n## is defined as a linear map ##D: C^{\infty}(\mathbb{R}^n) \to \mathbb{R}## which for ##f,g \in...
46. ### Mechanics question derivations terminal velocity (quadratic case)

hi so basically my question is derivation of derivation of the quadratic case of the terminal velocity I have attached the pictures in picture 1 I don't know I think the book has error in his derivation @ equation 2.4.13 shouldn't it be + instead of negative ?? & in second picture their...
47. ### Some basic derivations need improvement

The traditional Schrodinger equation is written as Laplacian ψ+constant(E-V)ψ=0 the above equation derivation is based on the formation of hydrogen after the combination of proton and electron. E is said to be total energy and V potential. energy of hydrogen atom. On solving the value of E...
48. ### Derivations of the series expansions

Are there derivations of the taylor, Fourier and laurant series?
49. ### Proofs, derivations, or both? Feel I've learned math/physics wrong

I've recently come to the conclusion that I might have made some mistakes along the way. I'm going into my senior year of EE and something just doesn't feel right about my abilities. Over the last couple semesters, I've fallen into the "plug and chug" mode of solving problems. I have some issues...
50. ### Derivations for a Bending Moment and Angle from Koenig's Apparatus

Homework Statement I'm having ALOT of trouble figuring out how to derive these equations. I'll post up photos and the equations as follows. C is the Bending Moment Y is the Young's Modulus of the material r is the Radius of Curvature of the neutral surface I is the geometrical moment...