MATH 251 Lecture 21

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Quadratic Form Example

$Q(x,y)=2x^2+2xy+2y^2$

$\begin{array}{rl}A& =\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)\\ X& =\left(\begin{array}{c}x\\ y\end{array}\right)\\ Q& =X^{T}AX=\left(\begin{array}{cc}x& y\end{array}\right)\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\\ \end{array}$

1. Lagrange Multiplier Equations:
   $AX=\lambda X$; $x^2+y^2=1$
2. Find characteristic polynomial and eigenvalues:
   $\{\begin{array}{l}(2-\lambda )x+y=0\\ x+(2-\lambda )y=0\end{array}$
   $\left|\begin{array}{cc}2-\lambda & 1\\ 1& 2-\lambda \end{array}\right|=(2-\lambda {)}^2-1=0$
   $\lambda =1,3$
3. Classify Q: positive
4. Find Eigenvectors:
   $\begin{array}{rl}\lambda =1& \to \{\begin{array}{l}(2-1)x+y=0\\ x+(2-1)y=0\end{array}⟶{\overrightarrow{v}}_1=\left(\begin{array}{c}\frac{\sqrt{2}}{2}\\ -\frac{\sqrt{2}}{2}\end{array}\right)\\ \lambda =3& \to \{\begin{array}{l}(2-3)x+y=0\\ x+(2-3)y=0\end{array}⟶{\overrightarrow{v}}_2=\left(\begin{array}{c}\frac{\sqrt{2}}{2}\\ \frac{\sqrt{2}}{2}\end{array}\right)\end{array}$
5. Use $\left(\begin{array}{c}x\\ y\end{array}\right)=x^{\prime }{\overrightarrow{v}}_1+y^{\prime }{\overrightarrow{v}}_2$ and plug $x$ and $y$ into $Q(x,y)$ to get it in terms of $x^{\prime }$ and $y^{\prime }$
   $\left(\begin{array}{c}x\\ y\end{array}\right)=\frac{\sqrt{2}}{2}\left(\begin{array}{c}{x}^{\prime }+y^{\prime }\\ -x^{\prime }+y^{\prime }\end{array}\right)$
   Even faster: $Q(x^{\prime },y^{\prime })={\lambda }_1(x^{\prime }{)}^2+{\lambda }_2(y^{\prime }{)}^2=(x^{\prime }{)}^2+3(y^{\prime }{)}^2$

Integration Over General Regions

The mindset behind this...

Let $R\subseteq {\mathbb{ℝ}}^2$ (any 2D region) and $Q$ be a rectangle that encloses $R$

$\underset{R}{\iint }f(x,y)\mathrm{d}A=\underset{Q}{\iint }f(x,y)\cdot {\chi }_R(x,y)\mathrm{d}A$

Where ${\chi }_R(x,y)$ is the characteristic equation for $f$ It "turns on" when we are inside $R$ and "turns off" when we are outside of $R$.

Fubini/Iterated Integrals

Let $R$ be a rectangle bounded by $[a,b]\times [c,d]$

$\underset{R}{\iint }f(x,y)\mathrm{d}A=\underset{c}{\overset{d}{\int }}(\underset{a}{\overset{b}{\int }}f(x,y)\mathrm{d}x)\mathrm{d}y=\underset{a}{\overset{b}{\int }}(\underset{c}{\overset{d}{\int }}f(x,y)\mathrm{d}y)\mathrm{d}x$

Example

$\underset{[0,1]\times [0,2]}{\iint }xy\mathrm{d}A$

$\begin{array}{rl}\underset{[0,1]\times [0,2]}{\iint }xy\mathrm{d}A& =\underset{0}{\overset{2}{\int }}\left(\underset{0}{\overset{1}{\int }}xy\mathrm{d}x\right)\mathrm{d}y\\ & =\underset{0}{\overset{2}{\int }}\left({\frac{x^2}{2}y|}_0^1\right)\mathrm{d}y\\ & =\underset{0}{\overset{2}{\int }}\frac{1}{2}y\mathrm{d}y\\ & ={\frac{y^2}{4}|}_0^2\\ & =1\end{array}$

A more sinister example

Integrate $f(x,y)=x^2+y$ over the region bounded by $y=x$, $y=x^2$.

$\begin{array}{rl}\underset{R}{\iint }(x^2+y)\mathrm{d}A& =\underset{0}{\overset{1}{\int }}\left(\underset{y}{\overset{\sqrt{y}}{\int }}(x^2+y)\mathrm{d}x\right)\mathrm{d}y\\ & =\underset{0}{\overset{1}{\int }}\left({\frac{x^3}{3}+xy|}_y^{y^{\frac{1}{2}}}\right)\mathrm{d}y\\ & =\underset{0}{\overset{1}{\int }}(\frac{4y^{\frac{3}{2}}}{3}-y^2-\frac{y^3}{3})\mathrm{d}y\\ & =\dots \end{array}$

A Just-Plain-Evil Example

$f(x,y)={\mathrm{e}}^x+y$ bounded by $y=x^2$ and $y=x^4$

$\begin{array}{rl}\underset{R}{\iint }f(x,y)\mathrm{d}A& =\underset{-1}{\overset{1}{\int }}\underset{x^4}{\overset{x^2}{\int }}({\mathrm{e}}^x+y)\mathrm{d}y\mathrm{d}x\\ & =\underset{-1}{\overset{1}{\int }}(x^2{\mathrm{e}}^x+\frac{1}{2}{x}^4-x^4{\mathrm{e}}^x-\frac{x^8}{2})\mathrm{d}x\\ & =\dots \end{array}$

Another Example

Evaluate the integral of ${\mathrm{e}}^{x^2}$ bounded by $y=x$, $y=2x$, and $x=[0,1]$ $\underset{0}{\overset{1}{\int }}\underset{x}{\overset{2x}{\int }}{\mathrm{e}}^{x^2}\mathrm{d}y\mathrm{d}x$