$$ \omega(x)= \sum c_I(x) dx_I $$

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

$$ \omega(x)= \sum c_I(x) dx_I $$

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

\(\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

\(\def\Real{\mathbb{R}}

\)

- To generalize the notion of 1-forms, we need to develop some algebraic apparatus. It is called
*Exterior Calculus.* -
Consider a vector space \(V\cong \mathbb{R}^n\).

We say that the functional

$$ \omega:\underbrace{V\times \cdots \times V}_{k \text{ times}}\to \mathbb{R} $$

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

\)

- 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

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

\)

- 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

\(\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&0\\0&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 …

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&0\\0&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.

Alternatively, one can think of them as the linear operators

\[

A:U\to U^*, …

**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 \(\{1,x,x^2\}\), …

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