MATH 251 Lecture 19

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Quick Fact

When does 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}ax+by=0\\cx+dy=0\end{cases}} 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 A \cdot X=0} have a non-zero solution?

in 2 dimensions, there is a unique solution unless the vectors 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\langle a,b \right\rangle} 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 \left\langle c,d \right\rangle} are scalar multiples (parallel)

in 3 dimensions (3 equations w/r/t 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} , 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} , and $z$ ), there is a unique solution unless the 3 vectors are coplanar.

Check the area of the 2 vectors (2D) and volume of 3 vectors (3D) by taking the determinant of the matrix formed by the coefficient. If this value is 0, then there are non-zero solutions. If the determiniant is 0, then the only solution is the point (0,0).

Written Homework Sample Problem

Given $Q(x,y)=x^{2}+2xy+3y^{2}$ ,

Set up equations satisfied by relative extrema on the unit circle (sphere for 3D)
Subject to the condition $g(x,y)=x^{2}+y^{2}=1$ , set $\nabla Q=\lambda \nabla g$ and $g=1$

Part A

Set $\nabla Q=\lambda \nabla g$ and our constraint

${\begin{cases}x+y=\lambda x\\x+3y=\lambda y\\x^{2}+y^{2}=1\end{cases}}$

Part B

Find the characteristic polynomial and solve for $\lambda$ . Start by putting everything on the left side of the equation:

${\begin{cases}(1-\lambda )x+y=0\\x+(3-\lambda )y=0\end{cases}}$

Find the determinant of the matrix ${\begin{vmatrix}1-\lambda &1\\1&3-\lambda \end{vmatrix}}=0$

The determinant gives $(1-\lambda )(3-\lambda )-1=0$ ... our characteristic polynomial.

The solution(s) for $\lambda$ are:

$\lambda ={\frac {4\pm {\sqrt {16-4(1)(2)}}}{2(1)}}=2\pm {\sqrt {2}}$

These are our Lagrange multipliers.

Part C

The values of $\lambda$ (eigenvalues) are the extreme values. Classify $Q$ as positive, negative, mixed, or degenerate.

All values of $\lambda$ are positive, therefore $Q$ is positive.

Part D

Find the solutions $(x,y)$ (eigenvectors) by plugging $\lambda$ into the equations from 'Part A. In order for the solution to be non-zero, the solutions should be redundant, so you can choose one equation and plug it into that one.

${\begin{aligned}\lambda _{1}=2+{\sqrt {2}}&\longrightarrow {\frac {1}{\sqrt {4+2{\sqrt {2}}}}}\left\langle 1,1+{\sqrt {2}}\right\rangle \\\lambda _{1}=2-{\sqrt {2}}&\longrightarrow {\frac {1}{\sqrt {4+2{\sqrt {2}}}}}\left\langle 1+{\sqrt {2}},-1\right\rangle \end{aligned}}$

Note: The lines given by plugging 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} in will always be perpendicular

Part E?

Diagonalize the formula by removing the mixed terms (i.e., the 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 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 Q(x',y') = \lambda_1(x')^2 + \lambda_2(y')^2}

Multidimensional Integration

Given 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 f(x,y)} defined over a rectangle 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=\left[a,b\right] \times \left[c,d\right] = \left\{(x,y) ~|~ a\le x \le b, c \le y \le d \right\}} , define 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 \iint_R f(x,y) \, \mathrm{d}A}

Back to Calculus 2, the integral 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_a^b f(x)\, \mathrm{d}x} represents

Area under a curve 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 f(x)}
Total mass of a rod 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 f(x)} is the linear density function
Probability density of a continuous random variable
The sum of an infinite family of numbers defined continuously in terms of 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}

In 3 Dimensions, the multidimensional integral 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 \iint_R f(x,y) \, \mathrm{d}A} represents

The volume under a function
Total mass in a given volume with mass density of 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 f(x,y)}
Probability density of two continuous random variables
Still an infinite sum, but determined continuously by two variables 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} 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 y} .

Quick Definition

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 \iint_R f(x,y) \, \mathrm{d}A = \lim_{|P|\to 0} \sum_{i,j=1}^N f(x_i,y_j) \Delta x_i \Delta y_j}

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 \Delta x_i \Delta y_j} represents the area of a small rectangle near 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_i, y_j)} , called a partition (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 P} )

Example

Calculate 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 \iint_R (x-y^2) \, \mathrm{d}A} , 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 R=\left[0,2\right] \times \left[1,3\right]} .

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_1^3 \int_0^2 (x-y^2) \, \mathrm{d}x \,\mathrm{d}y}

Notation is to use Fubini's Theorem: slice the function according to a single variable (by holding the other constant; 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 y} )

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} \int_0^2 (x-y^2) \, \mathrm{d}x &= \left. \frac{1}{2}x^2-xy^2 \right|_0^2\\ \int_1^3 (2-2y^2) \,\mathrm{d}y &= \left. 2y-\frac{2}{3}y^3 \right|_1^3 = 4-\frac{2}{3}26 \end{align}}