MATH 251 Lecture 21

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Quadratic Form Example

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} A &= \begin{pmatrix}2&1\\1&2\end{pmatrix} \\ X &= \begin{pmatrix}x\\y\end{pmatrix} \\ Q &= X^T A X = \begin{pmatrix}x&y\end{pmatrix} \begin{pmatrix}2&1\\1&2\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix} \\ \end{align}}

  1. Lagrange Multiplier Equations:
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AX = \lambda X} ; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2+y^2=1}
  2. Find characteristic polynomial and eigenvalues:
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} (2-\lambda)x+y=0\\x+(2-\lambda)y = 0\end{cases}}
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{vmatrix}2-\lambda & 1 \\ 1 & 2-\lambda\end{vmatrix}=(2-\lambda)^2-1=0}
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda = 1,3}
  3. Classify Q: positive
  4. Find Eigenvectors:
    Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\lambda =1&\rightarrow {\begin{cases}(2-1)x+y=0\\x+(2-1)y=0\end{cases}}\quad \longrightarrow \quad {\vec {v}}_{1}={\begin{pmatrix}{\frac {\sqrt {2}}{2}}\\-{\frac {\sqrt {2}}{2}}\end{pmatrix}}\\\lambda =3&\rightarrow {\begin{cases}(2-3)x+y=0\\x+(2-3)y=0\end{cases}}\quad \longrightarrow \quad {\vec {v}}_{2}={\begin{pmatrix}{\frac {\sqrt {2}}{2}}\\{\frac {\sqrt {2}}{2}}\end{pmatrix}}\end{aligned}}}
  5. Use and plug and into to get it in terms of and

    Even faster:


Integration Over General Regions

The mindset behind this...

Let (any 2D region) and be a rectangle that encloses

Where is the characteristic equation for It "turns on" when we are inside Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} and "turns off" when we are outside of .

Fubini/Iterated Integrals

Let be a rectangle bounded by

Example


A more sinister example

Integrate Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x,y)=x^{2}+y} over the region bounded by , .

A Just-Plain-Evil Example

bounded by and

Another Example

Evaluate the integral of bounded by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=x} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=2x} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=[0,1]} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^1 \int_x^{2x} \mathrm{e}^{x^2} \, \mathrm{d}y \, \mathrm{d}x}