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\centerline {\bf CALCULUS 223}
\centerline {\sl Final Exam Study Sheet}
\vskip .5in

\boxit{\bf Derivatives and Partials}

$$\nabla f\left(x,y,z\right) = \pmatrix{f_x \cr f_y \cr f_z \cr}$$

{\bf Maxima/Minima}

{
\parindent .4in
\item{i)} $f$ has a {\bf local maximum} at $(a,b)$ if $f_{xx},f_{yy} < 0$ and $f_{xx}f_{yy} - f_{xy}^2 > 0$ at $(a,b)$;
\item{ii)} $f$ has a {\bf local minimum} at $(a,b)$ if $f_{xx},f_{yy} > 0$ and $f_{xx}f_{yy} - f_{xy}^2 > 0$ at $(a,b)$;
\item{iii)} $f$ has a {\bf saddle point} at $f_{xx}f_{yy} - f_{xy}^2 < 0$ at $(a,b)$;
\item{iv)} The test is {\it inconclusive} at $(a,b)$ if $f_{xx}f_{yy} - f_{xy}^2 = 0${} at $(a,b)$.  Use different method.
}

\medskip
{\bf Tangent Plane} at point $p$ when:
$$\nabla f|_p \cdot \pmatrix{\Delta x \cr \Delta y \cr \Delta z \cr} = 0$$

{\bf LaGrange Muliplier}
$$\nabla f = \lambda \nabla g \hbox{\ and\ } g(x,y,z) = 0$$



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\boxit{\bf Multiple Integrals}

$$A_R = \int \int_R r\,dr\,d\theta = \int \int_R \,dy\,dx$$

$$\int_0^{2\pi} \int_0^\pi \int_0^1 \,\rho^2 \sin(\phi)\,dr\,d\phi\,d\theta$$

$$\int_0^{2\pi} \int_{r_i}^{r_o} \int_{z_a}^{z_b} \,r\,dz\,dr\,d\theta$$

$$\int_{y_a}^{y_b} \int_{x_a}^{x_b} \int_{z_a}^{z_b}\,dz\,dx\,dy$$

{\bf Polar}
$$x = r\cos(\theta)  \hskip .5in
y = r\sin(\theta)$$

{\bf Spherical}
$$x = \rho\sin(\phi)\cos(\theta)  \hskip .4in
y = \rho\cos(\phi)\sin(\theta) \hskip .4in
z = \rho\cos(\phi)$$
$$\rho^2 = x^2 + y^2 + z^2$$

{\bf Change of Variables $x,y$ to $u,v$}

$$J(u,v) = \left\|\matrix{
  {\partial x \over \partial u }   {\partial x \over \partial v}\cr
  {\partial y \over \partial u}   {\partial y \over \partial v}\cr}\right\| = {\partial(x,y)\over\partial(u,v)}$$

{\bf Mass and Moment}
$$\hbox{\bf Density\ } = \delta(x,y)$$
$$\hbox{\bf Mass\ }M = \int\int\delta(x,y)\,dA$$
{\bf First Moments\ }
$$M_x = \int\int y\delta(x,y)\,dA   \hskip .5in
M_y = \int\int x\delta(x,y)\,dA$$

{\bf Center of Mass}
$$\bar x = {M_y \over M} \hskip .4in \bar y = {M_x \over M}$$

{\bf Moments of Inertia (second moments)}
$$\hbox{About the $x$ axis\ }I_x = \int \int y^2\delta(x,y)\,dA$$
$$\hbox{About the $y$ axis\ }I_y = \int \int x^2\delta(x,y)\,dA$$
$$\hbox{About the origin\ }I_0 = \int \int \left(x^2 + y^2\right)\delta(x,y)\,dA = I_x + I_y$$
$$\hbox{About line $L$\ } I_L = \int \int r^2(x,y)\delta(x,y)\,dA  \hskip .3in \hbox{where $r(x,y)$ is the distance from $L$ to $(x,y)$}$$

{\bf Radii of Gyration}
$$\hbox{About the $x$ axis\ }R_x = \sqrt{{I_x \over M}}$$
$$\hbox{About the $y$ axis\ }R_y = \sqrt{{I_y \over M}}$$
$$\hbox{About the origin\ }R_0 = \sqrt{{I_0 \over M}}$$


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\boxit{\bf Line Integrals}

{\bf Line Integral} {\sl Find mass of string.}

$$\int F(s)\,ds = \int F(s) \sqrt{\left(x'\right)^2+\left(y'\right)^2+\left(z'\right)^2}\,dt$$

{\bf Circulation} {\sl Work through vector field.}
$$\oint M\,dx + N\,dy =
\int \int_R \left({\partial N \over \partial x} - {\partial M \over \partial y}\right)\,dx\,dy =
\int\vec F \cdot \,d\vec r =
\int\vec F \cdot \vec T \,ds $$
$$d\vec r = \nabla \vec r\,dt$$
$$F = \pmatrix{
  M \cr
  N \cr }$$

{\bf Circulation Density}
$$\hbox{\bf Curl\ }  = {\partial N \over \partial x} - {\partial M \over \partial y}$$

{\bf Flux}
$$  \oint M\,dy - N\,dx =
   \int \left( M {dy \over dt} - N {dx \over dt} \right) \,dt =
   \int\int_R \left({\partial M \over \partial x} + {\partial N \over \partial y}\right)\,dx\,dy =
   \int_a^b \vec F \cdot \left({\partial y \over \partial t} \vec i - {\partial x \over \partial t} \vec j\right)\,dt$$

{\bf Flux Density}
$$ {\partial M \over \partial x} + {\partial N \over \partial y}$$

{\bf Green's Theorem}
$$\oint_CM\,dy - N\,dx = \int\int_R\left({\partial M \over \partial x} + {\partial N \over \partial y}\right)\,dx\,dy$$

{\bf Green's Theorem Area Formula}
$$\hbox{Area of\ }R = {1 \over 2}\oint_C x\,dy - y\,dx$$


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\boxit{\bf First Order Differential Equations}

{\bf Seperable\ }{\sl Form} $P(x)dx = Q(y)dy$
{
\parindent .8in
\item{1.} Integrate.
}

{\bf Exact\ }{\sl Form} $Mdx + Ndy = 0$ when $M_y = N_x$
{
\parindent .8in
\item{1.} $\phi = \int M\,dx = f + h(y)$
\item{2.} $h'(y) = N - \phi_y$ \hskip .1in because \hskip .1in $\phi_y = f_y + h'(y)$
\item{3.} $h(y) = \int h'(y)\,dy$
\item{4.} $\phi = f + h(y)$
}

{\bf Linear\ }{\sl Form} $y' + p(x)y = q(x)$
{
\parindent .8in
\item{1.} $m(x) = e^{\int p(x)\,dx}$
\item{2.} $\int (my)'\,dx = \int m(x)q(x)\,dx$
\item{3.} $y = {f + C \over m(x)}$
}

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\boxit{\bf Second Order Differential Equations}

{\bf Homogeneous Linear\ }{\sl Form} $y'' + ay' + b = 0$
{
\parindent .8in
\item{1.} $r^2 + ar + b = 0$ \hskip .1 in find $r$.
\item{2.} $r_1$, $r_2$ Real, and $r_1 \ne r_2$ \hskip .2in $y = C_1e^{r_1x}+C_2e^{r_2x}$
\item{3.} $r_1$, $r_2$ Real, and $r_1 = r_2$ \hskip .2in $y = (C_1x + C_2)e^{r_1x}$
\item{4.} $r_1$, $r_2$ Complex Conjugates, $\alpha \pm \beta i$ \hskip .2in $y = e^{\alpha x}(C_1\cos(\beta x)+ C_2\sin(\beta x))$
}

{\bf Nonhomogeneous Linear\ }{\sl Form} $y'' + ay' + b = f(x)$
{

Method 1
\parindent .8in
\item{1.} Find $y_h$.
\item{2.} Find $y_p$ in same form as $f$.
\itemitem {a.} $C$ try $A$ or $Ax$
\itemitem {b.} $x^2$ try $Ax^2 + Bx + C$
\itemitem {c.} $\cos(2x)$ try $A\cos(2x) + B\sin(2x)$
\item{3.} Get $y_p = f$
\item{4.} $y = y_h + y_p$
}

{

Method 2
\parindent .8in
\item{1.} Find $y_h$.
\item{2.} $y_p = v_1y_1 + v_2y_2$ \hskip .5in $v_1(x)$, $v_2(x)$ \hskip .5in (match form)
\item{2.} Solve \hskip .1in $v_1'y_1 + v_2'y_2 = 0$ and $v_1'y_1' + v_2'y_2' = f(x)$ for $v_1$, $v_2$.
\item{3.} $y = y_h + y_p$
}
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\bye