Vectors as a differential op and covectors as differenential

In summary, the conversation discusses different ways of defining tangent vectors on a manifold. One definition is through equivalence classes of curves, while another is as derivations that obey the Leibniz rule. Both definitions are equivalent and can be used to understand the tangent space at a point on a manifold. Additionally, the notation df is an invariant way to describe the Jacobian matrix of a function as a linear map between vector spaces.
  • #1
binbagsss
1,254
11
Hey,

I'm new to this , and I understand the derivation of the transition laws for overlapping regions of a manifold for covectors and vectors starting from thinking of them as a differential and a differential operator respectively, but I don't really have a clue where this comes from...

Any assistance or a point toward some good source, greatly appreciated !
 
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  • #2
There's a more intuitive definition of the tangent space at p that isn't used much, because it's harder to work with. By this definition, a tangent vector at p is an equivalence class of curves through p. The equivalence relation is defined using a coordinate system ##x:U\to\mathbb R^n## such that ##p\in U##. We say that two curves ##C:\mathbb R\to M## and ##B:\mathbb R\to M## such that ##C(0)=B(0)=p## are equivalent if
$$(x\circ C)'(0)=(x\circ B)'(0).$$ It turns out that this definition is independent of the chosen coordinate system. We can then define addition and scalar multiplication on the set of equivalence classes to turn it into a vector space. And then we can prove that the tangent space defined this way is isomorphic to the already familiar tangent space at p. The isomorphism is the map ##\phi## defined by
$$\phi([C])(f)=(f\circ C)'(0)$$ for all equivalence classes [C] and all smooth functions f.

There's more information in my posts in this thread and the one I linked to from there.

The fact that ##\big((\mathrm dx^1)_p,\dots,(\mathrm dx^n)_p\big)## is the dual of ##\big(\frac{\partial}{\partial x^1}\big|_p,\dots,\frac{\partial}{\partial x^n}\big|_p\big)## is a straighforward consequence of the definition of the ##\mathrm d## operation. The definition is ##(\mathrm df)_p(v)=v(f)## for all smooth ##f:M\to\mathbb R## and all ##v\in T_pM##, so we have
$$(\mathrm dx^i)_p\left(\frac{\partial}{\partial x^j}\bigg|_p\right) = \frac{\partial}{\partial x^j}\bigg|_p x^i =\delta^i_j.$$
 
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  • #3
Although Fredrik has explained it all, I thought this might be helpful as well.

In ordinary calculus in Euclidean space, vectors operate on functions by directional derivatives. If v is a vector, its action on a function,f, is

lim t ->0 (f( x + tv) - f(x))/t

Conversely one can define vectors at a point,x, as linear operators on functions that obey the Leibniz rule, that is : (v.fg) = (v.f)g(x) + (v.g)f(x)

One can show that such a "derivation" is equal to a directional derivative for some vector.

Similarly df(v) is is defined by the same limit. The way I learned it was that the Jacobian matrix of a function is a linear map between vector spaces. For instance, the gradient of a function is a linear map from n space to the real numbers.The notation df is the invariant way to describe the Jacobian.

As Fredrik explained, on a manifold one only has directional derivatives in a coordinate chart so some notion of equivalence under change of charts is needed. One way to define vectors invariantly without reference to coordinate charts is as equivalence classes of smooth curves at a point, another is as derivations.
 
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1. What is a vector as a differential operator?

A vector as a differential operator is a mathematical object that acts on a function to produce a new function. It represents a direction and magnitude in which the function is being differentiated.

2. How are covectors different from vectors as differentials?

Covectors are dual to vectors and can be thought of as "row vectors" while vectors are "column vectors." They act on vectors to produce a scalar value, while vectors act on functions to produce new functions.

3. How are vectors and covectors used in differential geometry?

In differential geometry, vectors and covectors are used to describe the tangent and cotangent spaces at each point of a manifold. They are also used to define the metric tensor, which measures distances and angles on a manifold.

4. Can vectors and covectors be represented as matrices?

Yes, vectors and covectors can be represented as matrices. Vectors are typically represented as column matrices, while covectors are represented as row matrices. The entries in these matrices correspond to the components of the vector or covector in a particular basis.

5. How are vectors and covectors related to the gradient and divergence operators?

Vectors and covectors are related to the gradient and divergence operators through the use of the gradient and divergence theorems. The gradient of a scalar function is a covector, while the divergence of a vector field is a scalar function. These relationships are important in the study of vector calculus and its applications.

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