# Archive | work

## March 28

• Consider the $$k$$-form $$\omega$$ defined at $$x\in \Omega\subset \mathbb{R}^n$$, given by the sum
$$\omega(x)= \sum c_I(x) dx_I$$
where the multi-index
$$I=\{1\leq i_1\leq i_2 \leq, \ldots, \leq i_k \leq n\}$$.
Each basis element
$$dx_I:= dx_{i_1} \wedge • ## Exterior Differential forms \(\def\pd{\partial} \def\Real{\mathbb{R}}$$

• We define a differential $$k$$-form as
$$\omega(x) = \sum_{I=\{i_1<i_2<\ldots<i_k\}} c_I(x) dx_{i_1}\wedge \cdots\wedge dx_{i_k},$$
an exterior form with coefficients depending on the positions.

(We will be using sometimes simplifying notation $$\xi_i = dx_i$$ or $$\eta_i=dy_i$$.)

• Pullback maps: Let

## Exterior Forms

$$\def\Real{\mathbb{R}}$$

1. To generalize the notion of 1-forms, we need to develop some algebraic apparatus. It is called Exterior Calculus.
2. Consider a vector space $$V\cong \mathbb{R}^n$$.
We say that the functional
$$\omega:\underbrace{V\times \cdots \times V}_{k \text{ times}}\to \mathbb{R}$$

## Calculus of Exterior and Differential Forms: Motivation

$$\def\pd{\partial}\def\Real{\mathbb{R}}$$

1. Let $$\Omega \subset \mathbb{R}^n$$ be some Euclidean domain, and consider
a curve $$\gamma:\underbrace{[0,1]}_{I} \to \Omega$$ given by
$$\gamma(t) = \begin{pmatrix} \gamma_1(t), & \ldots &, \gamma_n(t) \end{pmatrix}.$$Given a real-valued multivariate function $$f:\Omega\to \mathbb{R}$$ we’ll try to

## 3.14 Vector Analysis And Classical Identities in 3D

$$\def\pd{\partial}\def\Real{\mathbb{R}}$$

1. There are two main types of products between vectors in $$\mathbb{R}^3$$:
The inner/scalar/dot product
$$A\cdot B = A_x+B_x + A_yB_y + A_z Bz \in \mathbb{R}$$
is commutative, distributive, and homogenenous.
The vector (cross) product:
$$A\times ## 2.28 operators in Hermitian spaces and their spectra. $$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}\def\Field{\mathbb{F}}\def\Fun{\mathbf{Fun}}\def\e{\mathbf{e}} \def\f{\mathbf{f}}\def\bv{\mathbf{v}}\def\i{\mathbf{i}} \def\eye{\left(\begin{array}{cc}1&amp;0\\0&amp;1\end{array}\right)} \def\bra#1{\langle #1|}\def\ket#1{|#1\rangle}\def\j{\mathbf{j}}\def\dim{\mathrm{dim}} \def\ker{\mathbf{ker}}\def\im{\mathbf{im}} \def\tr{\mathrm{tr\,}} \def\braket#1#2{\langle #1|#2\rangle}$$ 1. Given a bilinear form $$Q(\cdot,\cdot)$$, or, equivalently, a mapping $$Q:U\to U^*$$, one can bake easily new bilinear forms from linear operators $$U\to U$$: just take $$Q_A(u,v):=Q(u,Av)$$. This opens … ## midterm problems and solutions here.… ## solutions to 2.14 Exercise: Let $$V$$ be the space of real polynomial functions of degree at most $$3$$. Consider the quadratic form $$q_1:V \to \mathbb{R}$$ given by$$ q_1(f):=|f(-1)|^2+|f(0)|^2+|f(1)|^2. $$Is this form positive definite? Consider another form $$q:V\to \mathbb{R}$$ given by$$ …

$$\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}\def\Field{\mathbb{F}}\def\Fun{\mathbf{Fun}}\def\e{\mathbf{e}} \def\f{\mathbf{f}}\def\bv{\mathbf{v}}\def\i{\mathbf{i}} \def\eye{\left(\begin{array}{cc}1&amp;0\\0&amp;1\end{array}\right)} \def\bra#1{\langle #1|}\def\ket#1{|#1\rangle}\def\j{\mathbf{j}}\def\dim{\mathrm{dim}} \def\ker{\mathbf{ker}}\def\im{\mathbf{im}} \def\tr{\mathrm{tr\,}} \def\braket#1#2{\langle #1|#2\rangle}$$
1. Bilinear forms are functions $$Q:U\times U\to k$$ that depend on each of the arguments linearly.
Exercise: consider the mapping that takes a quadratic polynomial $$q$$ to its values at $$0,1,2$$ and $$3$$. Find the normal form of this operator.
Solution: let $$V$$ denote the space of quadratic polynomials with the standard basis $$\{1,x,x^2\}$$, …