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:
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{\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.
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...
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...
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...
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...
[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...
The effect of temperature change in Le Chatelier's Principle is given by the equation $$ \log \left(\frac{\mathrm{K}_{2}}{\mathrm{~K}_{1}}\right)=\frac{\Delta \mathrm{H}}{2.303 \mathrm{R}}\left[\frac{1}{\mathrm{~T}_{1}}-\frac{1}{\mathrm{~T}_{2}}\right] $$. How to derive $$ \log...
Hello! So what I've tried to tackle this problem is derive the equation,set it equal to zero,find a value for v and than put it in the second derivation.So when I derive this I get $$ \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.
##...
Greetings
I'm currently designing a lab-script intended for comprehension and use at an undergraduate level. I was extremely frustrated during my undergrad to be dealing with a plethora of uninspired and dull experiments so I decided to take a slightly unconventional (or extremely conventional...
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...
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...
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...
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...
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...
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.)
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...
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...
My text of physics, Gettys's, proves that the magnetic field on the axis of a solenoid, in whose loops, of linear density ##n## (i.e. there are ##n## loops per length unit), a current of intensity ##I## flows, has the same direction as the loops' moment of magnetic dipole and magnitude ##\mu_0...
Homework Statement
I've been looking at examples of motion derivations for my class, and it's honestly just very confusing. I heard Dynamics should prep you for this but I must have had a very poor course because we never had to understand geometry and physics to this degree...
Homework...
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 ?
Hi. I would like to ask a simple question. Here is the link of the file I study on. Immediately before the formula 4.9 for Biot number. Lc=V/As but I cannot understand it and I think it is not clear enough. How it appears, for what the word "characteristic" stands for, for example a pipe? For...
Homework Statement
A straight circular plastic cylinder of length L and radius R (where
R ≪ L)
is irradiated with a beam of protons so that there is a total excess charge Q distributed uniformly throughout the cylinder. Find the electric field inside the cylinder, a distance r from the center...
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'...
I do not get some points of this proof about the time derivative of a unit vector $\hat{u}$ (costant magnitude) which is following a precession motion. The picture is the following.
I want to prove that $$\frac{d\hat{u}}{dt}=\vec{\Omega}\wedge \hat{u}.$$
I'm ok with almost all the proof except...
Homework Statement
Derive Cholesky Decomposition for a 3x3 matrix
Homework Equations
IN: S is Real matrix with dimensions 3x3 and is Symmetric and semi-definite
Out: L is a Real matrix with dimensions 3x3 such that
S=L*L^t
L is lower-triangular
The Attempt at a Solution
We learned this in...
I am reading John M. Lee's book: Introduction to Smooth Manifolds ...
I am focused on Chapter 3: Tangent Vectors ...
I need some help in fully understanding Lee's conversation on directional derivatives and derivations ... ... (see Lee's conversation/discussion posted below ... ... )
Lee...
Hello,
Please take a look at this handbook of derivatives and integrals:
http://myhandbook.info/form_diff.html
http://integral-table.com/downloads/single-page-integral-table.pdf
I would appreciate it if someone could point me in the direction of exemplary books that derive these...
I am reading the book Mathematical Logic by Ian Chiswell and Wilfred Hodges ... and am currently focused on Chapter 3: Propositional Logic and, in particular, Section 3.4: Propositional Natural Deduction ...
I need help with understanding an aspect of Example 3.4.3 which reads as...
Hello! I understand what specific heats are and how to derive them. I just feel that I'm missing a little something in the methodology.
Consider the 1st law of thermodynamics and the definition of enthalpy:
1) dU = δQ -δW = δQ - PdV
2) H = Q - VP
For the derivation of CV, dV = 0 and the...
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...
(Change in momentum is the area under a force against time curve.)
(Force is the time derivative of momentum.)
Using separation of varibles, you get Fdt=dp. Integrate both sides, you get that the integral of Force with respect to time is equal to p. This seems to imply that p, momentum, is...
I don't understand how some terms are derived.
How did the last term of 3-47 originate?
How did 3-49 get so many terms from just one term in 3-41?
Why integrate from V to infinite? That is not intuitive.
Thas a functions are unusual because the absolute values of U,H,S cannot be computed...
Hi Guys, new poster here.
I am currently doing a practical report on Uniform Circular Motion, where we had to swing a rubber stopper around attached to a length of string and mass.
I have to do a write up, including the background theory. I have searched everywhere but I have found no clear...
Homework Statement
I am wondering if anyone knows of a document that shows derivations of specific values in Physics? For example, I need to find μs and am given the distance traveled and time (1.2 km in 17 sec).
Homework Equations
I know the equation for μs, Ffriction=μs⋅FN.
The Attempt...
I have been studying potential steps and barriers as well as reflection and transmission coefficients and how to derive them. Most of it makes sense to me except for one thing:
As we know, the normal Schrodinger equation is:
(-ħ2/2m) (∂2Ψ/∂x2) + v(x)Ψ = EΨ
For a step potential however, my...
Hey,
I just had the chance to extract the gravitational lensing caused by a massive point using Fermat's principle.
I was wondering though, is there any other way to do that?
Also is the light's time delation induced by the "refraction index" n (Saphiro delay) connected to "gravitational time...
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...
Hello Everyone,
I have read many derivations of Einstein field equations (done one myself), but none of them explain why the constant term should have a $$c^4$$ in the denominator. the 8πG term can be obtained from Poisson's equation, but how does c^4 pop up? Most of the books just derive it...
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...
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...
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...
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...
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...
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...
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...
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...
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...