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:


{\displaystyle D(ab)=aD(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. H

    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. S

    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. Q

    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. BWV

    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. K

    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. E

    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...
  7. Huzaifa

    Chemistry Effect of temperature change in Le Chatelier's Principle

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  8. A

    Determining the maximum braking power using derivations

    Hello! So what I've tried to tackle this problem is derive the equation,set it equal to zero,find a value for v and than put it in the second derivation.So when I derive this I get $$ \frac{v^2+9-v *(2v)}{v^4+8v^2+16)} $$ Now if i set that equal 0 and try to find a value for v I get this. ##...
  9. R

    Understanding kT/hc in Derivations

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  10. JacobPhys

    Experimental Derivations of Pi in Physics

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  11. Math Amateur

    MHB The Space of All Derivations at a point p .... Tu, Theorem 2.2 .... ....

    I am reading Loring W.Tu's book: "An Introduction to Manifolds" (Second Edition) ... I need help in order to fully understand Theorem 2.2 and the remarks after the theorem ...Theorem 2.2 and the remarks after the theorem read as follows: My questions on the above text from Tu are as follows...
  12. Math Amateur

    MHB Tangent Vectors in R^n as Derivations ....

    I am reading Loring W.Tu's book: "An Introduction to Manifolds" (Second Edition) ... I need help in order to fully understand Tu's section on tangent vectors in \mathbb{R}^n as derivations... In his section on tangent vectors in \mathbb{R}^n as derivations, Tu writes the following: In the above...
  13. wirefree

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    G'day, Would anybody else agree that derivations of key results in physics texts - school or college level - proceed in a manner that betrays no compunctions in the mind of the author about fore-knowledge of the pre-known form of the final expression? I have a strong conviction that the great...
  14. Wrichik Basu

    Studying Is learning derivations of formulae always very important?

    I understand that it is useful to learn and remember the derivations of formulae in most cases. However, I tend to forget the derivations of several formulae, especially those in optics and dynamics. For a moment, let's forget the examinations. I wish to pursue higher studies in applied...
  15. N

    A Entropy and derivations - is my logic faulty?

    It is assumed that entropy increases in the universe. However, the fluid and acceleration equations are derived assuming that. TdS=dE+pdV where dQ = TdS. But dQ is usually set equal to zero to derive these equations. Hence since T is non zero, dS should be zero and so there would be no...
  16. S

    Studying How do i learn derivations and diagrams?

    I study ncert physics class 12. Is there any special way you rote learned all the diagrams and derivations? (for example they ask in board exam that explain the working and principle of cyclotron and draw its diagram. Very hard to remember.)
  17. D

    Which path has the shortest time?

    Guys I have the following homework problem to solve: There are 2 given points in a plane. If we take a point-like object with mass m and take it to the "higher" point what path should it go on to reach the other point in the shortest possible time. Only gravitational force affects our point-like...
  18. D

    A Questions about Einstein's 1916 GR Paper: Answers Needed

    Hello everyone. I was reading Einsteins 1916 original paper on GR, the "The foundation of the general theory of relativity". There are some derivation that he did but I didn't quite understand. It would be nice if someone can give me some direction or some guidance on it. Here is the link to...
  19. DavideGenoa

    I Properties of ideal solenoid: postulates or derivations?

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  20. W

    Can someone explain this derivation?

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  21. mertcan

    Quantum Quantum Theory Books for Understanding Derivations

    hi, I really want to dig valuable things out of quantum theory, also I have a big eagerness to see the derivation of formulas to understand the logic of this topic. Could you recommend me some nice books which may meet my needs I expressed at the beginning ?
  22. M

    Some simple heat transfer formula derivations and questions

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  23. yeezyseason3

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  24. A

    MHB Simple to understand derivations similar to the Taylor Series

    That I don't even know in which forum to post this questions shows my gaping lack of mathematics knowledge. I've just learned the derivation of the Taylor series. I'm slapping myself on the head as it's so mind-bogglingly simple, but I never learned it. The Taylor series was just 'maths magic'...
  25. S

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  26. BobTheLawyer

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  27. Math Amateur

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  28. samgrace

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  29. Math Amateur

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  30. Kori Smith

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  31. VoteSaxon

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    Homework Statement Using the special relativity formulae p = mv / [1 - (v/c)2] E2 = p2c2 + m2c4 derive linear relations between: (i) momentum and mass; (ii) energy and mass; (iii) energy and momentum, which involve only c, c2, β = v/c, and γ (= 1/sqrt(1 - β2)) The attempt at a solution I am...
  32. Kyuutoryuu

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  33. S

    Question about derivations of thermodynamic properties

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  34. A

    Uniform Circular Motion Background theory?

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  35. B

    Derivations of specific values in Physics

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  36. S

    Derivations for Schrodinger's equations for potential step

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  37. ChrisVer

    Gravitational Lensing Derivations - Is There Another Way?

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  38. S

    Learning derivations versus solving problems

    Hi, I understand when I study a physics topic, it's very important that I solve as many problems as I can possibly can and it's also very important to internalise all the techniques used to solve problems in that topic. However, I was wondering if the same applies for the derivation of formulae...
  39. C

    Derivations of Einstein field equations

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  40. topsquark

    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...
  41. B

    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...
  42. Z

    Should I memorize all the derivations in QM?

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  43. B

    Derivations - What's Acceptable?

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  44. C

    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...
  45. G

    Mechanics question derivations terminal velocity (quadratic case)

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  46. D

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  47. B

    Derivations of the series expansions

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  48. D

    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...
  49. N

    Derivations for a Bending Moment and Angle from Koenig's Apparatus

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