Archive | ece 493

RSS feed for this section

problems for review

  1. Consider the language \(L\) consisting of all words in alphabet a,b,c having no subwords aaa. Find the generating function
    f(x)=\sum |L_n| z^n,
    where \(L_n\) is the set of words of lengths \(n\) in \(L\). Estimate how fast \(\#
Read full story Comments { 0 }

Stokes and relatives

  • The (general) Stokes Theorem: this unique theorem puts
    Green’s/Gauss/Stokes Theorems (from intermediate calculus)
    under one umbrella. Let \(C\) be a polyhedron with boundary \(\partial C\). Then
    $$ {\int_{\partial C} \omega = \int_C d\omega }$$
  • Example: integrate the form \(\omega =
  • Read full story Comments { 0 }

    solutions for problems on vector analysis and differential forms


    Assume that 4 vectors in \(\mathbb{R}^3\) satisfy \(A+B+C+D\)=0.
    $$ A\times B-B\times C+C \times D-D\times A$$

    Solution: By the anti-symmetry \(-B\times C = C\times B\)
    and \(-D\times A = A\times D\) we can rewrite the expression as
    $$ …

    Read full story Comments { 0 }

    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
  • Read full story Comments { 0 }

    Exterior Differential forms


    • 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
    Read full story Comments { 0 }

    Exterior Forms


    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
      $$ \omega:\underbrace{V\times \cdots \times V}_{k \text{ times}}\to \mathbb{R} $$
    Read full story Comments { 0 }

    Calculus of Exterior and Differential Forms: Motivation


    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
    Read full story Comments { 0 }

    3.14 Vector Analysis And Classical Identities in 3D


    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
    Read full story Comments { 0 }

    2.28 operators in Hermitian spaces and their spectra.

    \def\bra#1{\langle #1|}\def\ket#1{|#1\rangle}\def\j{\mathbf{j}}\def\dim{\mathrm{dim}}
    \def\braket#1#2{\langle #1|#2\rangle}

    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 …

    Read full story Comments { 0 }


    problems and solutions here.…

    Read full story Comments { 0 }