Derivations vs Directional derivatives

In summary, a derivation at a point is a linear map from a Cartesian coordinate space to a vector space. It is more general than a directional derivative, which is a special case of a derivation. A derivation in differential geometry is the Lie derivative of a vector field.
  • #1
center o bass
560
2
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 C^\infty(\mathbb{R}^n)## that satisfies

##D(fg) = f(a) Dg + g(a) Df.##

On the other hand, some books just stick to directional derivatives, so I wondered: what is the virtue of introducing derivations?

Can someone give an example of something that is a derivation, but not a directional derivative?
 
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  • #2
center o bass said:
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 C^\infty(\mathbb{R}^n)## that satisfies

##D(fg) = f(a) Dg + g(a) Df.##

On the other hand, some books just stick to directional derivatives, so I wondered: what is the virtue of introducing derivations?

Can someone give an example of something that is a derivation, but not a directional derivative?
Derivations are a more general term. One definition of the tangent space at a point ##p## on a manifold ##M## is that the tangent space ##T_pM## is the vector space of all derivations at ##p##. However, I can't think of anywhere where you can't think of elements of the tangent space as directional derivatives.

In Lie algebras, the map ##\operatorname{ad}_x:\mathfrak{g}\to\mathfrak{g}## that takes ##y\in\mathfrak{g}## to ##[x,y]\in\mathfrak{g}## (The ##[\cdot,\cdot]## is called the bracket. It's the multiplication operation on ##\mathfrak{g}##.) is a derivation, but it isn't a directional derivative.
 
  • #3
If v is a derivation at p, and ##x:U\mapsto\mathbb R^n## is a coordinate system such that ##p\in U##, then there exist real numbers ##v^1,\dots,v^n## such that
$$v=\sum_{k=1}^n v^k\frac{\partial}{\partial x^k}\!\bigg|_p.$$ If this right-hand side is what you consider a directional derivative, then this means that every derivation is a directional derivative.

Not sure if ##\mathrm{ad}_x## should really be considered a derivation, since it's ##\mathfrak g##-valued rather than ##\mathbb R##-valued.
 
  • #4
Fredrik said:
Not sure if ##\mathrm{ad}_x## should really be considered a derivation, since it's ##\mathfrak g##-valued rather than ##\mathbb R##-valued.

It's still called a derivation: http://en.wikipedia.org/wiki/Derivation_(differential_algebra )

Anyway, the Lie bracket can be seen as an actual derivative if you use the notion of Lie derivatives, so its not too farfetched to call it a derivation.
 
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  • #5
micromass said:
It's still called a derivation: http://en.wikipedia.org/wiki/Derivation_(differential_algebra))

Anyway, the Lie bracket can be seen as an actual derivative if you use the notion of Lie derivatives, so its not too farfetched to call it a derivation.
(...ERMAGERD Micro's back! :biggrin:)

Micro is correct. There is a construction in differential geometry known as the Lie derivative, which can be applied to various tensor fields. We define the Lie derivative of a vector field ##Y## with ##X## to be ##\mathcal{L}_X(Y)=[X,Y]##.

To generalize, for any given noncommutative algebra ##A##, the map ##D_x: y\mapsto xy-yx## is a derivation.

Proof: ##D_x(yz)=xyz-yzx=xyz-yxz+yxz-yzx=D_x(y)z+yD_x(z)##.
 

FAQ: Derivations vs Directional derivatives

What is the difference between a derivation and a directional derivative?

A derivation is a mathematical concept that represents the rate of change of a function at a specific point. It is a generalization of the derivative in multi-dimensional spaces. A directional derivative, on the other hand, is the rate of change of a function in a specific direction at a specific point. It is a special case of a derivation.

How do you calculate a derivation?

To calculate a derivation, you first find the partial derivatives of the function with respect to each variable. Then, you use these partial derivatives to construct a linear combination, which represents the direction of the derivation. Finally, you evaluate this linear combination at the given point to calculate the derivation.

Can a directional derivative be negative?

Yes, a directional derivative can be negative. The sign of a directional derivative depends on the angle between the direction of the derivative and the gradient of the function at that point. If the angle is obtuse, the directional derivative will be negative.

What is the significance of directional derivatives in real-life applications?

Directional derivatives have many applications in real-life, especially in fields such as physics, engineering, and economics. For example, in physics, directional derivatives are used to calculate the velocity and acceleration of moving objects. In economics, they are used to analyze the sensitivity of a function to changes in certain variables.

How are derivations and directional derivatives related?

Derivations and directional derivatives are closely related concepts. A directional derivative is a special case of a derivation, and they both represent the rate of change of a function at a specific point. However, derivations are more general and can be used in higher dimensions, while directional derivatives are limited to one specific direction.

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