Inculcation: Analysis in finite-dimensional vector spaces

parent:: inculcation

• A lot of these material is present in Napkin but it does not give enough page-length to fully grasp the whole topic, maybe?
• shifrin-multivariable-mathematics does a good job at the calculus
• Analysis by Herbert Amann and Joachim Escher Volume I or Rudin or Apostol contains the analysis

Derivatives, differential forms and beyond - YouTube

This is me explaining tensors, derivatives and differential forms, focusing mostly on motivation of the philosophy and also looking for no-brain symbolic calculation as a summary.

Link to original

total and directional derivative of functions §

Total derivative of function

$$f:U\subseteq V\to W$$

between finite-dim real normed vector spaces at a point $p\in U$ is supposed to be a linear map

$$D_{p}f:V\to W$$

that “approximates” $f$ near $p$. So if $D_{p}f$ is a rotation (as a linear map), then

$$f:U\to f(U)$$

should “look” like a rotation “near $p$“.

measurable functions and their integrals §

AKA ”measure theory“.

differential forms, their integration and exterior derivative §

We look at the (familiar) structure of

$$\begin{array}{ccccccc}{\mathcal{C}}^{\infty }(U)& \stackrel{\mathrm{grad}}{\to }& {\mathrm{Vec}}^{\infty }(U)& \stackrel{\mathrm{curl}}{\to }& {\mathrm{Vec}}^{\infty }(U)& \stackrel{\mathrm{div}}{\to }& {\mathcal{C}}^{\infty }(U)\\ & & \mathrm{curl}\circ \mathrm{grad}=0& & & & \\ & & & & \mathrm{div}\circ \mathrm{curl}=0& & \end{array}$$

where $U\subseteq {\mathbb{R}}^3$.

Writing this in a different notation: for ${\mathcal{C}}^{\infty }(U)\stackrel{\mathrm{grad}}{\to }{\mathrm{Vec}}^{\infty }(U)\stackrel{\mathrm{curl}}{\to }{\mathrm{Vec}}^{\infty }(U)$

$$\begin{array}{ccccc}f& \stackrel{\mathrm{d}}{\mapsto }& \mathrm{d}f=\underset{\text{this looks like grad}}{\underbrace{\sum _i\frac{\partial f}{\partial x_i}\mathrm{d}{x}_i}}& & \\ & & \omega ={\sum }_i{\omega }_i\mathrm{d}{x}_i& \stackrel{\mathrm{d}}{\mapsto }& \mathrm{d}\omega =\underset{\text{this looks like curl}}{\underbrace{\sum _i\mathrm{d}{\omega }_i\wedge \mathrm{d}{x}_i}}\end{array}$$

and then continuing for ${\mathrm{Vec}}^{\infty }(U)\stackrel{\mathrm{div}}{\to }{\mathcal{C}}^{\infty }(U)$

$$\begin{array}{cccccccc}& & & & \mathrm{d}\omega =\underset{\text{this looks like curl}}{\underbrace{\sum _i\mathrm{d}{\omega }_i\wedge \mathrm{d}{x}_i}}& & & \\ & & & & \alpha ={\sum }_{i>j}{\alpha }_{ij}\mathrm{d}{x}_i\wedge \mathrm{d}{x}_j& \stackrel{\mathrm{d}}{\mapsto }& \mathrm{d}\alpha =\underset{\text{this looks like div (check!)}}{\underbrace{\sum _{i>j}\mathrm{d}{\alpha }_{ij}\wedge \mathrm{d}{x}_i\wedge \mathrm{d}{x}_j}}& \\ & & & & & & \beta ={\beta }_{123}\mathrm{d}{x}_1\wedge \mathrm{d}{x}_2\wedge \mathrm{d}{x}_3& \stackrel{\mathrm{d}}{\mapsto }0\end{array}$$

where we understand that the “wedge” $\wedge$ works like the cross product

$$\mathrm{d}{x}_i\wedge \mathrm{d}{x}_j=-\mathrm{d}{x}_j\wedge \mathrm{d}{x}_i$$

It is easily seen

$$\mathrm{d}(\mathrm{d}(-))=0$$

doesn’t matter what’s inside the $(-)$, a real function $f$ or the objects $\omega$, $\alpha$ or $\beta$ (called 0,1,2,3-forms respectively) when their components are differentiable functions.

We see this “exterior derivative”

$$\mathrm{d}$$

thus “unifies” grad, curl and div and generalizes because it may be defined as

$$\text{d}\omega =\mathrm{d}\left(\sum {\omega }_I\mathrm{d}{x}_I\right):=\mathrm{d}{\omega }_I\wedge \mathrm{d}{x}_I$$

in any dimension.

But what are these

$$\mathrm{d}{x}_i$$

mean?

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Integrating differential forms on chains (generalizing line, surface, volume integrals) is very simple

$$\int \omega$$

Here we have the fundamental theorem of calculus but generalized:

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The de Rham complex

$${\Omega }^k\stackrel{\mathrm{d}}{\to }{\Omega }^{k+1}$$

begs the question, when is a closed form $\mathrm{d}\omega =0$ exact (there exists $g$ such that $\mathrm{d}g=\omega$?)

The Poincare lemma says locally, any closed form is exact. Fine, if you’re happy with it. But there are closed forms whose integrals are not zero.

Hence they are not exact. Then when are closed forms exact, globally?

But plot twist! This question is purely topological!

things that requires a metric §

Hodge duals

algebraic structures that naturally came up §

• Vector space, linear transformations
• Tensor product of vector spaces
• Algebras of a vector space: Tensor algebra, symmetric algebra, exterior algebra
• Exact sequences, cohomology

vector fields and their flows §

The relevant material is in inculcation-odes

vector fields and ODE dictionary §

What we do is, write a differential equation like

$${\dot{x}}_i=f_i(x_1,x_2,\dots ,x_n)$$

for all $1\le i\le n$ so we have $n$ equations for $n$ variables $x_i$ and make it even more compact by

$$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$$

where $\mathbf{f}:\mathcal{U}\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ is a vector field on the open domain $\mathcal{U}$.

This gives us a geometric pov on ODEs in ${\mathbb{R}}^n$, and we have a

solving differential equations	analysis and geometry of vector fields
an equation $\dot{x}=\mathbf{f}(x)$	a vector field $\mathbf{f}:\mathcal{U}\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$
solutions of the equation $\begin{array}{rl}\dot{x}(t)& =\mathbf{f}(x(t))\\ x(0)& =x_0\end{array}$	integral curves of the vector field $t\mapsto {\Phi }_t^{\mathbf{f}}(x_0)$
how solutions depend on initial conditions $x_0\mapsto x(t)$	flows of the vector field $x_0\mapsto {\Phi }_t^{\mathbf{f}}(x_0)$
conserved quantities	integrals of the vector field
(linearly) decoupling the differential equation $\dot{z_i}=f_i(z_i)$	(linear) coordinate transformation such that $\mathbf{f}={\sum }_i{f}_i(z_i){\hat{z}}_i$

This is a standard geometric interpretation.

We may convert ordinary differential equation of any order to first order by taking enough independent variables and defining them to be higher derivatives.

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in complex vector spaces §

We may mimic everything above to complex vector spaces with a norm!

This for example leads to differential forms of the form

$$f(z)\mathrm{d}z$$

where $\mathrm{d}z:=\mathrm{d}x+i\mathrm{d}y$.

This starts the study of complex analysis (well, if we consider complex analytic $f$ :).

next: generalize! §

Transclude of inculcation-linear-constructions#bggpgh