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}}
- 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} - 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} - Classify Q: positive
- 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}}} - 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}