Sergei Yakovenko's blog: on Math and Teaching

Sunday, May 6, 2018

Lecture 7, May 6, 2018

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 4:05
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Integration of differential 1-forms. Differential k-forms and their integration

Integration of a 1-form over a smooth curve: use pull-back or approximate by the Riemann sums? Independence of the parameterization.

$\displaystyle \int_\gamma\mathrm df=f(\text{end})-f(\text{start})$.

Integral of $y\,\mathrm dx$ along a smooth closed curve: area bounded by the curve.

Multilinear forms on a linear n-space. Tensor product.

Symmetric and antisymmetric multilinear forms. Symmetrization and alternation. Determinant. Wedge product. Algebraic properties of the wedge product, action by adjoint operators.

Differential k-forms and their integration along smooth k-dimensional “pieces” (images of a cube).

Lecture 6, Apr 29, 2018

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 3:55
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Differential 1-forms

Vectors vs. covectors, covector field = differential 1-form. Functoriality: how to pull them back by smooth maps. Examples: differentials of smooth functions. Module of differential 1-forms over the structure algebra of smooth functions. Lie derivation along a flow of vector field.

De Rham and Cech cohomology of smooth manifolds

Using the exterior differential $d$ on smooth differential forms and the fact that $d^2=0$, we define the de Rham cohomology with real coefficients $H^k_{\mathrm dR}(M,\mathbb R)$ as the quotient space of closed $k$-forms by exact $k$-forms. This is a global invariant of a manifold $M$ (for non-compact manifolds we may also consider a version for compactly supported forms, which yields different results).

De Rham cohomology can be computed using the Poincare lemma. If $\mathfrak U=\{U_i\}$ is an open covering of $M$ such that all opens sets and all their non-empty finite intersections are topologically trivial (homeomorphic to open balls), then for any closed form $\omega\in\Omega^k(M)$ one can construct its primitives $\xi_i\in\Omega^{k-1}(U_i)$ such that $\mathrm d\xi_i=\omega$ in $U_i$. The $(k-1)$th forms $\xi_i$ may disagree on the intersections $U_{ij}=U_i\cap U_j$, but one can attempt to twist them by suitable closed forms $\mathrm d\phi_i$. The corresponding system of $(k-2)$-forms $\{\phi_i\}$ satisfies certain linear conditions on pairwise intersections; to satisfy these conditions one has to look for forms on triple intersections etc.

This construction gives rise to the notion of the Cech cohomology defined via systems of linear algebraic equations and reduces computation of the de Rham cohomology to a problem from linear algebra, determined by the combinatorics of the pairwise and multiple intersections of the sets $U_i$. In particular, one can conclude that the de Rham cohomology of compact manifolds is finite-dimensional.

An ultra-concise set of notes is available here (I hope to return and expand this text). The notion of Cech cohomology is further elaborated here.

NB. The class was shorter than usual because of the Hanukka lighting ceremony.

חנוכה שמח and Happy New Year, С наступающим Новым годом!

Exterior derivation

The differential $\mathrm df$ of a smooth function $f$ is in a sense container which conceals all directional derivatives $L_Xf=\left\langle\mathrm df,X\right\rangle$ along all directions, and dependence on $X$ is linear.

If we consider the directional Lie derivative $L_X\omega$ for a form $\omega\in\Omega^k(M)$ of degree $k\ge 1$, then simple computations show that $L_{fX}\omega$ is no longer equal to $f\cdot L_X\omega$. However, one can “correct” the Lie derivative in such a way that the result will depend on $X$ linearly. For instance, if $\omega\in\Omega^1(M)$ and $X$ is a vector field, we can define the form $\eta_X\in\Omega^1(M)$ by the identity $\eta_X=L_X\omega-\mathrm d\left\langle\omega,X\right\rangle$ and show that the 2-form $\eta(X,Y)=\left\langle\eta_X,Y\right\rangle$ is indeed bilinear antisymmetric.

The 2-form $\eta$ is called the exterior derivative of $\omega$ and denoted $\mathrm d\omega\in\Omega^2(M)$. The correspondence $\mathrm d\colon\Omega^1(M)\to\Omega^2(M)$ is an $\mathbb R$-linear operator which satisfies the Leibniz rule $\mathrm d(f\omega)=f\,\mathrm d\omega+(\mathrm df)\land \omega$ and $\mathrm d^2 f=0$ for any function $f\in\Omega^0(M)$.

It turns out that this exterior derivation can be extended to all $k$-forms preserving the above properties and is a nice (algebraically) derivation of the graded exterior algebra $\Omega^\bullet(M)=\bigoplus_{k=0}^n\Omega^k(M)$.

The lecture notes are available here.

Multilinear antisymmetric forms and differential forms on manifolds

We discussed the module of differential 1-forms dual to the module of smooth vector fields on a manifold. Differential 1-forms are generated by differentials of smooth functions and as such can be pulled back by smooth maps.

The “raison d’être” of differential 1-forms is to be integrated over smooth curves in the manifold, the result being dependent only on the orientation of the curve and not on its specific parametrization.

At the second hour we discussed the notion of forms of higher degree, which required to introduce the Grassman algebra on the dual space $T^*$ to an abstract finite-dimensional linear space $T\simeq\mathbb R^n$. The Grassmann (exterior) algebra is a mathematical miracle that was discovered by a quest for unusual and unknown, with only slight “motivations” from outside.

The day ended up with the definition of the differential $k$-forms and their functoriality (i.e., in what direction and how they are carried by smooth maps between manifolds).

The lecture notes are available here.

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