MATH 251 Lecture 22

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Exam on Friday

Written Homework:

1. Calculating determinants
2. Polar gradient: 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 f = f_x \hat\imath + f_y \hat\jmath = f_r \hat{r} + \frac{f_{\theta}}{r} \hat{\theta}}
3. Classifying Quadratic Forms: 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 = ax^2+2bxy+cy^2 \Rightarrow \begin{pmatrix}a&b\\c&d\end{pmatrix}} (positive, negative, mixed, degenerate)

Integrals in Polar Coordinates

• convert region to polar coordinates: $R_{x,y}$ to $R_{r,\theta }$
• convert function to polar coordinates: $x=r\cos {\theta }$, $y=r\sin {\theta }$
• convert differentials to polar coordinates: $\mathrm {d} x\rightarrow \mathrm {d} r$, $\mathrm {d} y\rightarrow r\mathrm {d} \theta$

Example

Region in first quadrant of $x^{2}+y^{2}=2$ excluding the part bounded by $x^{2}+(y-1)^{2}=1$

$\int _{0}^{1}\int _{\sqrt {\dots }}^{\sqrt {\dots }}y\,\mathrm {d} x\,\mathrm {d} y$

In Polar Coordinates:

Formulas

${\begin{aligned}C_{1}:~x^{2}+y^{2}-2y+1=1&~\Rightarrow ~r=2\sin {\theta }\\C_{2}:~x^{2}+y^{2}=2&~\Rightarrow ~r={\sqrt {2}}\end{aligned}}$

Region

${\begin{aligned}0\leq \theta \leq \arcsin {\left({\frac {r}{2}}\right)}\\0\leq r\leq {\sqrt {2}}\end{aligned}}$

${\begin{aligned}2\sin {\theta }\leq r\leq {\sqrt {2}}0\leq \theta \leq {\frac {\pi }{4}}\end{aligned}}$

$\int _{0}^{\frac {\pi }{4}}\int _{2\sin {\theta }}^{\sqrt {2}}r^{2}\sin {\theta }\,\mathrm {d} r\,\mathrm {d} \theta =\ldots$

Mass / Center of Mass / Moment of Inertia

Mass density function $\rho (x,y)$, meaning that for a small area $\left(x^{*},y^{*}\right)$, the mass of that region is 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 \rho(x^*,y^*)\cdot A} .

• The total mass in the region 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} is 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 \rho(x,y) \mathrm{d}A}
• The center of mass (not necessarily the middle of region) is 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 (\bar{x},\bar{y}) = (M_x,M_y)} , 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 \bar{x}} is the mean value 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} .
  • 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 \bar{x} = \frac{1}{\mathrm{mass}} \iint_R x \cdot \rho(x,y) \,\mathrm{d}A}
  • 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 \bar{y} = \frac{1}{\mathrm{mass}} \iint_R y \cdot \rho(x,y) \,\mathrm{d}A}

Example

Use region as half-washer with outer radius 2, inner radius 1, and θ between 0 and π

Density = Constant = 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 \rho_0}

Mass = 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 \rho_0 \,\mathrm{d}A = \rho_0 \cdot \mathrm{Area} = \frac{3\pi}{2} \rho_0}

Center of Mass:

• 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 \bar{x} = \iint_R x\rho_0 \,\mathrm{d}A = 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 \bar{y} = \iint_R y\rho_0 \,\mathrm{d}A = \frac{28}{9\pi}}