MATH 251 Lecture 21

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Quadratic Form Example

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

${\begin{aligned}A&={\begin{pmatrix}2&1\\1&2\end{pmatrix}}\\X&={\begin{pmatrix}x\\y\end{pmatrix}}\\Q&=X^{T}AX={\begin{pmatrix}x&y\end{pmatrix}}{\begin{pmatrix}2&1\\1&2\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}\\\end{aligned}}$

1. Lagrange Multiplier Equations:
   $AX=\lambda X$; $x^{2}+y^{2}=1$
2. Find characteristic polynomial and eigenvalues:
   ${\begin{cases}(2-\lambda )x+y=0\\x+(2-\lambda )y=0\end{cases}}$
   ${\begin{vmatrix}2-\lambda &1\\1&2-\lambda \end{vmatrix}}=(2-\lambda )^{2}-1=0$
   $\lambda =1,3$
3. Classify Q: positive
4. Find Eigenvectors:
   ${\begin{aligned}\lambda =1&\rightarrow {\begin{cases}(2-1)x+y=0\\x+(2-1)y=0\end{cases}}\quad \longrightarrow \quad {\vec {v}}_{1}={\begin{pmatrix}{\frac {\sqrt {2}}{2}}\\-{\frac {\sqrt {2}}{2}}\end{pmatrix}}\\\lambda =3&\rightarrow {\begin{cases}(2-3)x+y=0\\x+(2-3)y=0\end{cases}}\quad \longrightarrow \quad {\vec {v}}_{2}={\begin{pmatrix}{\frac {\sqrt {2}}{2}}\\{\frac {\sqrt {2}}{2}}\end{pmatrix}}\end{aligned}}$
5. Use ${\begin{pmatrix}x\\y\end{pmatrix}}=x'{\vec {v}}_{1}+y'{\vec {v}}_{2}$ and plug $x$ and $y$ into $Q(x,y)$ to get it in terms of $x'$ and $y'$
   ${\begin{pmatrix}x\\y\end{pmatrix}}={\frac {\sqrt {2}}{2}}{\begin{pmatrix}x'+y'\\-x'+y'\end{pmatrix}}$
   Even faster: $Q(x',y')=\lambda _{1}(x')^{2}+\lambda _{2}(y')^{2}=(x')^{2}+3(y')^{2}$

Integration Over General Regions

The mindset behind this...

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

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

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 $[a,b]\times [c,d]$

$\iint _{R}f(x,y)\,\mathrm {d} A=\int _{c}^{d}\left(\int _{a}^{b}f(x,y)\,\mathrm {d} x\right)\,\mathrm {d} y=\int _{a}^{b}\left(\int _{c}^{d}f(x,y)\,\mathrm {d} y\right)\,\mathrm {d} x$

Example

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

${\begin{aligned}\iint _{[0,1]\times [0,2]}xy\,\mathrm {d} A&=\int _{0}^{2}\left(\int _{0}^{1}xy\,\mathrm {d} x\right)\,\mathrm {d} y\\&=\int _{0}^{2}\left(\left.{\frac {x^{2}}{2}}y\right|_{0}^{1}\right)\,\mathrm {d} y\\&=\int _{0}^{2}{\frac {1}{2}}y\,\mathrm {d} y\\&=\left.{\frac {y^{2}}{4}}\right|_{0}^{2}\\&=1\end{aligned}}$

A more sinister example

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

${\begin{aligned}\iint _{R}(x^{2}+y)\,\mathrm {d} A&=\int _{0}^{1}\left(\int _{y}^{\sqrt {y}}\left(x^{2}+y\right)\,\mathrm {d} x\right)\,\mathrm {d} y\\&=\int _{0}^{1}\left(\left.{\frac {x^{3}}{3}}+xy\right|_{y}^{y^{\frac {1}{2}}}\right)\,\mathrm {d} y\\&=\int _{0}^{1}\left({\frac {4y^{\frac {3}{2}}}{3}}-y^{2}-{\frac {y^{3}}{3}}\right)\,\mathrm {d} y\\&=\ldots \end{aligned}}$

A Just-Plain-Evil Example

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

${\begin{aligned}\iint _{R}f(x,y)\,\mathrm {d} A&=\int _{-1}^{1}\int _{x^{4}}^{x^{2}}(\mathrm {e} ^{x}+y)\,\mathrm {d} y\,\mathrm {d} x\\&=\int _{-1}^{1}\left(x^{2}\mathrm {e} ^{x}+{\frac {1}{2}}x^{4}-x^{4}\mathrm {e} ^{x}-{\frac {x^{8}}{2}}\right)\,\mathrm {d} x\\&=\ldots \end{aligned}}$

Another Example

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