MATH 251 Lecture 18

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Quadratic Forms

$Q(x,y)=ax^{2}+2bxy+cy^{2}$

For the second derivative test, we replace $f(x,y)$ with its second order Taylor polynomial:

$f(x_{0}+\Delta x,y_{0}+\Delta y)\approx f(x_{0},y_{0})+{\frac {1}{2}}\left(f_{xx}(x_{0},y_{0})\cdot (\Delta x)^{2}+2f_{xy}(x_{0},y_{0})\Delta x\Delta y+f_{yy}(x_{0},y_{0})\cdot (\Delta y)^{2}\right)$

Classification

Examine the values of $Q(x,y)$ for $f(x,y)$ on the unit circle $x^{2}+y^{2}=1$. Find two relative extrema $\lambda _{1}$ and $\lambda _{2}$

if all positive

a paraboloid whose vertex represents a minimum

if all negative

a paraboloid whose vertex represents a maximum

if mixed

a saddle surface

if some value on degenerate

a parabolic cylinder representing a line of minimum or maximum values

Lagrange Multipliers

Useful for finding extrema along a curve or surface domain (just the problem we were discussing earlier).

Find relative extrema of $f(x,y)$ for $(x,y)$ constrained by $g(x,y)=C$.

$g(x,y)=C$ represents a level curve of the function $g$

The general idea is to follow the curve $g(x,y)=C$ along the direction of $\nabla f$ to find the maximum value. At all extrema points, $\nabla f$ and $\nabla g$ will be parallel. Since they are parallel, one will be a scalar multiple of the other.

We're left with the following 3 equations with unknown values $x$, $y$, and $\lambda$.

${\begin{cases}{\frac {\partial f}{\partial x}}=\lambda {\frac {\partial g}{\partial x}}\\{\frac {\partial f}{\partial y}}=\lambda {\frac {\partial g}{\partial y}}\\g(x,y)=C\end{cases}}$

Example

Going back to our previous line of thought, find the extrema of $2xy=f(x,y)=Q(x,y)$ on $g(x,y)=x^{2}+y^{2}=1$:

${\begin{aligned}\nabla Q=\left\langle 2y,2x\right\rangle \\\nabla g=\left\langle 2x,2y\right\rangle \end{aligned}}$

Thus our three equations are 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} 2y = \lambda 2x \\ 2x = \lambda 2y \\ x^2+y^2 = 1 \end{cases}}

Solving these equations gives 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 = \pm 1} OR 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} .

• If 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} , then 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=0} , and (0,0) is not on our unit circle. We need a nonzero solution.
• If 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} , then 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=y} , and the points are 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 \left( \tfrac{\sqrt{2}}{2}, \tfrac{\sqrt{2}}{2} \right),\ \left( -\tfrac{\sqrt{2}}{2}, -\tfrac{\sqrt{2}}{2} \right)}
• If 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} , then 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=y} , and the points are 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 \left( \tfrac{\sqrt{2}}{2}, -\tfrac{\sqrt{2}}{2} \right),\ \left( -\tfrac{\sqrt{2}}{2}, \tfrac{\sqrt{2}}{2} \right)}

Plugging all solutions for a respective 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} into 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 Q(x,y)} gives the same 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} value... Coincidence? I think not!

For Quadratic forms, the Lagrange multipliers 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} are actually equal to the relative extrema. The values for 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} are called eigenvalues, and the characteristic polynomial we solved for 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} was 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^2-1=0}

In this case, 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 Q} is mixed

Matrix Form

For 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 Q(x,y) = ax^2+2bxy+cy^2} ,

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 Q(x,y) = \left[ A\begin{pmatrix}x\\y\end{pmatrix} \right] \cdot \begin{pmatrix}x\\y\end{pmatrix}} , where 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 A=\begin{pmatrix}a&b\\b&c\end{pmatrix}}

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 \nabla Q = \left\langle 2ax + 2by, 2bx + 2cy \right\rangle} .

The Lagrange multiplier method states that 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 \nabla Q = \lambda \nabla g}

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} 2ax + 2by &= \lambda 2x \\ 2bx + 2cy &= \lambda 2y \\ \end{align}}

This can be simplified down to 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 A\begin{pmatrix}x\\y\end{pmatrix} = \lambda \begin{pmatrix}x\\y\end{pmatrix}}

For our previous problem 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 Q=2xy} , 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 A = \begin{pmatrix}0&1\\1&0\end{pmatrix}} , which produces the same two equations for lambda with our constraint curve. <math>\begin{cases} y = \lambda x \\ x = \lambda y \\ x^2+y^2 = 1 \end{cases}