Derivation (differential algebra)

In mathematics, a derivation is a function on an algebra that 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(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 Der_K(A). The collection of K-derivations of A into an A-module M is denoted by Der_K(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. That is,

[FG,N]=[F,N]G+F[G,N],

where $[\cdot ,N]$ is the commutator with respect to $N$ . 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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Transcription

Welcome to the presentation on derivatives. I think you're going to find that this is when math starts to become a lot more fun than it was just a few topics ago. Well let's get started with our derivatives. I know it sounds very complicated. Well, in general, if I have a straight line-- let me see if I can draw a straight line properly-- if I had a straight line-- that's my coordinate axes, which aren't straight-- this is a straight line. But when I have a straight line like that, and I ask you to find the slope-- I think you already know how to do this-- it's just the change in y divided by the change in x. If I wanted to find the slope-- really I mean the slope is the same, because it is a straight line, the slope is the same across the whole line, but if I want to find the slope at any point in this line, what I would do is I would pick a point x-- say I'd pick this point. We'd pick a different color-- I'd take this point, I'd pick this point-- it's pretty arbitrary, I could pick any two points, and I would figure out what the change in y is-- this is the change in y, delta y, that's just another way of saying change in y-- and this is the change in x. delta x. And we figured out that the slope is defined really as change in y divided by change in x. And another way of saying that is delta-- it's that triangle-- delta y divided by delta x. Very straightforward. Now what happens, though, if we're not dealing with a straight line? Let me see if I have space to draw that. Another coordinate axes. Still pretty messy, but I think you'll get the point. Now let's say, instead of just a regular line like this, this follows the standard y equals mx plus b. Let's just say I had the curve y equals x squared. Let me draw it in a different color. So y equals x squared looks something like this. It's a curve, you're probably pretty familiar with it by now. And what I'm going to ask you is, what is the slope of this curve? And think about that. What does it mean to take the slope of a curve now? Well, in this line, the slope was the same throughout the whole line. But if you look at this curve, doesn't the slope change, right? Here it's almost flat, and it gets steeper steeper steeper steeper steeper until gets pretty steep. And if you go really far out, it gets extremely steep. So you're probably saying, well, how do you figure out the slope of a curve whose slope keeps changing? Well there is no slope for the entire curve. For a line, there is a slope for the entire line, because the slope never changes. But what we could try to do is figure out what the slope is at a given point. And the slope at a given point would be the same as the slope of a tangent line. For example-- let me pick a green-- the slope at this point right here would be the same as the slope of this line. Right? Because this line is tangent to it. So it just touches that curve, and at that exact point, they would have-- this blue curve, y equals x squared, would have the same slope as this green line. But if we go to a point back here, even though this is a really badly drawn graph, the slope would be something like this. The tangent slope. The slope would be a negative slope, and here it's a positive slope, but if we took a point here, the slope would be even more positive. So how are we going to figure this out? How are we going to figure out what the slope is at any point along the curve y equals x squared? That's where the derivative comes into use, and now for the first time you'll actually see why a limit is actually a useful concept. So let me try to redraw the curve. OK, I'll draw my axes, that's the y-axis-- I'll just do it in the first quadrant-- and this is-- I really have to find a better tool to do my-- this is x coordinate, and then let me draw my curve in yellow. So y equals x squared looks something like this. I'm really concentrating to draw this at least decently good. OK. So let's say we want to find the slope at this point. Let's call this point a. At this point, x equals a. And of course this is f of a. So what we could try to do is, we could try to find the slope of a secant line. A line between-- we take another point, say, somewhat close, to this point on the graph, let's say here, and if we could figure out the slope of this line, it would be a bit of an approximation of the slope of the curve exactly at this point. So let me draw that secant line. Something like that. Secant line looks something like that. And let's say that this point right here is a plus h, where this distance is just h, this is a plus h, we're just going h away from a, and then this point right here is f of a plus h. My pen is malfunctioning. So this would be an approximation for what the slope is at this point. And the closer that h gets, the closer this point gets to this point, the better our approximation is going to be, all the way to the point that if we could actually get the slope where h equals 0, that would actually be the slope, the instantaneous slope, at that point in the curve. But how can we figure out what the slope is when h equals 0? So right now, we're saying that the slope between these two points, it would be the change in y, so what's the change in y? It's this, so that this point right here is-- the x coordinate is-- my thing just keeps messing up-- the x coordinate is a plus h, and the y coordinate is f of a plus h. And this point right here, the coordinate is a and f of a. So if we just use the standard slope formula, like before, we would say change in y over change in x. Well, what's the change in y? It's f of a plus h-- this y coordinate minus this y coordinate-- minus f of a over the change in x. Well that change in x is this x coordinate, a plus h, minus this x coordinate, minus a. And of course this a and this a cancel out. So it's f of a plus h, minus f of a, all over h. This is just the slope of this secant line. And if we want to get the slope of the tangent line, we would just have to find what happens as h gets smaller and smaller and smaller. And I think you know where I'm going. Really, we just want to, if we want to find the slope of this tangent line, we just have to find the limit of this value as h approaches 0. And then, as h approaches 0, this secant line is going to get closer and closer to the slope of the tangent line. And then we'll know the exact slope at the instantaneous point along the curve. And actually, it turns out that this is the definition of the derivative. And the derivative is nothing more than the slope of a curve at an exact point. And this is super useful, because for the first time, everything we've talked about to this point is the slope of a line. But now we can take any continuous curve, or most continuous curves, and find the slope of that curve at an exact point. So now that I've given you the definition of what a derivative is, and maybe hopefully a little bit of intuition, in the next presentation I'm going to use this definition to actually apply it to some functions, like x squared and others, and give you some more problems. I'll see you in the next presentation

Properties

If A is a K-algebra, for K a ring, and $D : A \to A$ is a K-derivation, then

If A has a unit 1, then D(1) = D(1²) = 2D(1), so that D(1) = 0. Thus by K-linearity, D(k) = 0 for all $k \in K$ .
If A is commutative, D(x²) = xD(x) + D(x)x = 2xD(x), and D(xⁿ) = nxⁿ⁻¹D(x), by the Leibniz rule.
More generally, for any $x 1, x 2, \dots, x n \in A$ , it follows by induction that
$D(x_{1}x_{2}\cdots x_{n})=\sum _{i}x_{1}\cdots x_{i-1}D(x_{i})x_{i+1}\cdots x_{n}$

which is

{\textstyle \sum _{i}D(x_{i})\prod _{j\neq i}x_{j}}

if for all

i

D (x i)

commutes with

x_{1},x_{2},\ldots ,x_{i-1}

For n > 1, Dⁿ is not a derivation, instead satisfying a higher-order Leibniz rule:

D^{n}(uv)=\sum _{k=0}^{n}{\binom {n}{k}}\cdot D^{n-k}(u)\cdot D^{k}(v).

Moreover, if M is an A-bimodule, write

\operatorname {Der} _{K}(A,M)

for the set of K-derivations from A to M.

Der_K(A, M) is a module over K.
Der_K(A) is a Lie algebra with Lie bracket defined by the commutator:

[D_{1},D_{2}]=D_{1}\circ D_{2}-D_{2}\circ D_{1}.

since it is readily verified that the commutator of two derivations is again a derivation.

There is an A-module $Ω A / K$ (called the Kähler differentials) with a K-derivation $d : A \to Ω A / K$ through which any derivation $D : A \to M$ factors. That is, for any derivation D there is a A-module map $φ$ with

D:A{\stackrel {d}{\longrightarrow }}\Omega _{A/K}{\stackrel {\varphi }{\longrightarrow }}M

The correspondence

D\leftrightarrow \varphi

is an isomorphism of A-modules:

\operatorname {Der} _{K}(A,M)\simeq \operatorname {Hom} _{A}(\Omega _{A/K},M)

If $k \subset K$ is a subring, then A inherits a k-algebra structure, so there is an inclusion

\operatorname {Der} _{K}(A,M)\subset \operatorname {Der} _{k}(A,M),

since any K-derivation is a fortiori a k-derivation.

Graded derivations

Given a graded algebra A and a homogeneous linear map D of grade |D| on A, D is a homogeneous derivation if

{D(ab)=D(a)b+\varepsilon ^{|a||D|}aD(b)}

for every homogeneous element a and every element b of A for a commutator factor ε = ±1. A graded derivation is sum of homogeneous derivations with the same ε.

If ε = 1, this definition reduces to the usual case. If ε = −1, however, then

{D(ab)=D(a)b+(-1)^{|a|}aD(b)}

for odd |D|, and D is called an anti-derivation.

Examples of anti-derivations include the exterior derivative and the interior product acting on differential forms.

Graded derivations of superalgebras (i.e. Z₂-graded algebras) are often called superderivations.

Related notions

Hasse–Schmidt derivations are K-algebra homomorphisms

A\to A[[t]].

Composing further with the map which sends a formal power series $\sum a_{n}t^{n}$ to the coefficient $a_{1}$ gives a derivation.

References

Bourbaki, Nicolas (1989), Algebra I, Elements of mathematics, Springer-Verlag, ISBN 3-540-64243-9.
Eisenbud, David (1999), Commutative algebra with a view toward algebraic geometry (3rd. ed.), Springer-Verlag, ISBN 978-0-387-94269-8.
Matsumura, Hideyuki (1970), Commutative algebra, Mathematics lecture note series, W. A. Benjamin, ISBN 978-0-8053-7025-6.
Kolař, Ivan; Slovák, Jan; Michor, Peter W. (1993), Natural operations in differential geometry, Springer-Verlag.

This page was last edited on 17 October 2023, at 15:33

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