\documentstyle[]{article}
\begin{document}



\begin{center} \parbox{11cm}{
Checklist Week 1:  \\

\parbox{10cm}{
\fbox{I  Definitions} \\

   coordinates, vectors, sphere, dot product, cross product
   triple cross product, parallel vectors, orthogonal vectors,
   vector projection ${\rm P}_v(w)$ and the length of the 
   vector projection sometimes called scalar projection
}


\parbox{10cm}{
\fbox{II Facts}  \\

   $v.w = v_1 w_1 + v_2 w_2 + v_3 w_3  = |v| |w| \cos(\alpha)$ \\
   $|v \times w| = |v| |w| \sin(\alpha)$ is area of paralellogram \\
   $|u \cdot (v \times w)|$ is the volume of the parallelepiped
}

 
\parbox{10cm}{
\fbox{III Algorithms} \\

   Adding, subtracting and scaling vectors geometrically as well
   as algebraically. Completion of square.
   Compute dot, cross, triple products. 
   Find distance between points. Find vector
   orthogonal to two vectors, area of parallelogram, volume of a 
   parallelepiped spanned by three vectors. 
}


} \end{center} 


}



\begin{center} \parbox{11cm}{
Checklist Week 2:  \\

\parbox{10cm}{
\fbox{I  Definitions} \hfill
    \begin{tabular}{|c|c|} \hline
    Line & plane  \\ \hline
    $\frac{(x-x_0)}{a} = \frac{(y-y_0)}{b} = \frac{z-z_0}{c}$ & $ax + by + cz = d$ \\
    $\vec{x} = \vec{x}_0 + t \vec{v}$ & $\vec{x} = \vec{x}_0 + t \vec{v} + s \vec{w}$. \\ \hline
    \end{tabular}
    Plane and space curves $\vec{r}(t)$, 
    Velocity $\vec{r} \; '(t)$, Acceleration $\vec{r} \; ''(t)$,
    Position from velocity $\vec{r}(t) = \int_0^t \vec{r}'(s) \; ds + \vec{r}(0)$ and
    from acceleration.
    Unit tangent $\vec{T}(t)=\vec{r} \; '(t)/|\vec{r} \; '(t)|$,
    Unit normal $\vec{N}(t) = \vec{T}'(t)/|\vec{T}'(t)|$,
    Binormal vector $\vec{B}(t)=\vec{T}(t) \times \vec{N}(t)$. 
}


\parbox{10cm}{
\fbox{II Facts}  \\
     Plane $ax+by+cz=d$ has normal vector $\vec{n}=(a,b,c)$. 
     Line $\frac{(x-x_0)}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}$ contains
     $\vec{v}=(a,b,c)$,
     $\vec{r} \; '(t)$ is tangent to the curve,
     $\vec{v}=\vec{r} \; '$ then $\vec{r}=\int_0^t \vec{v} \; dt + \vec{c}$,
     arc length is independent of parametrization.
     $\vec{T},\vec{N},\vec{B}$ are perpendicular to each other. 
     Identities like $\frac{d}{dt} (\vec{v}(t) \cdot \vec{w}(t)) = \vec{v}'(t) \cdot \vec{w}(t)
     + \vec{v}(t) \cdot \vec{w} '(t)$.
}

 
\parbox{10cm}{
\fbox{III Algorithms} \\
    Using dot and cross product to find distances (point-plane,
    point-line, line-line, plane-plane).
    Geometric constructions like finding plane through $P,Q,R$. Intersection
    of two planes, intersection of line-plane, angles between lines and planes,
    switch from different descriptions of lines and planes. \\
    Compute $|\vec{r} \; '(t)|, \vec{T}(t)$ for curve $\vec{r}(t)$.
    Compute arc length of curve, arc length parametrization,
    Draw curves in the plane or in space. \\
    Match curves with their parametric equations. \\
    Solve free fall problems. 
}


} \end{center} 





\begin{center} \parbox{11cm}{
Checklist Week 3:  \\

\parbox{10cm}{
\fbox{I Definitions}  \\
    Curvature $\kappa(t) = |\vec{T}'(t)|/|\vec{r} \; '(t)|$ \\
    Domain and range of functions $f(x,y)$.
    Graph $G=\{ (x,y,f(x,y)) \}$. {\bf Intercepts}: intersections of $G$
    with coordinate axes. {\bf Traces}: intersections with
    coordinate planes. {\bf Generalized traces:} intersections with $\{ x=c \}$,
    $\{ y=c \}$  or $\{ z = c\}$.
    {\bf Quadrics:} Ellipsoid, Paraboloid, 1 and 2 sheeted Hyperboloid, Cylinder,
    Cone, Hyperbolic Paraboloid.  \\
}

\parbox{10cm}{
\fbox{II Facts}  \\
    The curvature of a circle of radius $r$ is $1/r$.\\
    Curvature is independent of parametrization. \\
    $\kappa(t) = \frac{ |r \; '(t) \times r \; ''(t)| }{ |r \; '(t)|^3 }$. \\
    Sets $g(x,y,z) = c$ describe {\bf surfaces}. \\
    Special cases: {\bf graphs} like $g(x,y,z)=z-\sin(x y)$,
    planes like $2x + 3y + 5z =7$ and {\bf quadrics} 
    like $x^2-2 y^2+3 z^2=4$.
}

\parbox{10cm}{
\fbox{III Algorithms} \\
    Know how to compute curvature (when formula is given). Know what
    curvature means. Find domain and range of functions of several variables.
    Sketch and match graphs of $f(x,y)$. Sketch and match quadrics. 
    Recognize conic sections (ellipses, hyperbola and parabola).
    Recognize quadrics using completion of squares (2 paraboloids, 2 hyperboloids, ellipsoids,
    or cone, cylinders)
}

} \end{center} 





\begin{center} \parbox{12cm}{
Checklist Week 4:  \\

\parbox{10cm}{
\fbox{I Definitions}  \\
    {\bf polar} $(x,y)=(r \cos(\theta), r \sin(\theta) )$.  \\
    {\bf cylindrical} $(x,y,z) = (r \cos(\theta), r \sin(\theta),z)$  \\
    {\bf spherical} $(x,y,z) = (\rho \cos(\theta) \sin(\phi), \rho \sin(\theta) \sin(\phi), \rho \cos(\phi))$  \\
    $g(r,\theta)=0$ {\bf polar curve}, especially $r = f(\theta)$, polar graphs. \\
    $g(r,\theta,z)=0$ {\bf "cylindrical surface"}, especially $r=f(z,\theta)$ or $r=f(z)$ {\bf surface of revolution}. \\
    $g(\rho,\theta,\phi)=0$ {\bf "spherical surface"} especially $\rho=f(\theta,\phi)$. \\
    $\vec{r}(u,v) = (x(u,v),y(u,v),z(u,v))$ parametrized surface. Fix one variable: grid curves. 
}

\parbox{10cm}{
\fbox{II Examples}  \\
   $x^2+y^2=r^2$ , $\vec{r}(t)=(r \cos(t) ,r \sin(t) )$. \\
   $x^2+y^2+z^2=\rho^2$,  $\vec{r}(\theta,\phi)=(\rho \cos(\theta) \sin(\phi),
                                                 \rho \sin(\theta) \sin(\phi),
                                                 \rho \cos(\phi))$. \\
   $a x + b y + c z = d$, $\vec{r}(u,v) = \vec{r}_0 + u \vec{v} + v \vec{w}, (a,b,c)=\vec{v} \times \vec{w}$. \\
   $r(\theta,z)=g(z)$ , $\vec{r}(u,v)=(g(v) \cos(u), g(v) \sin(u),v)$. \\
   $g(x,y,z) = z-f(x,y) = 0$ , $\vec{r}(u,v) = (u,v,f(u,v))$. \\
}

\parbox{10cm}{
\fbox{III Algorithms} \\
    plot curves and surfaces from implicit or parametric equation. \\
    match curves and surfaces with equations. \\
    parametrize curves and surfaces.  \\
    parametric description $\vec{r}(u,v)$  $\leftrightarrow$ implicit description $g(x,y,z)=0$ 
    for planes, sphere, graphs, surfaces of revolution. \\
    translate from and into polar coordinates. \\
    translate from and into spherical or cylindrical coordinates. \\
}

} \end{center} 





\begin{center} \parbox{12cm}{
Checklist Week 5:  \\

\parbox{10cm}{
\fbox{I Definitions}  \\
$f_x(x,y) = \frac{\partial}{\partial x} f(x,y)$  {\bf partial derivative} \\
{\bf partial differential equation} PDE like $F(f,f_x,f_t,f_{xx},f_{tt})=0$ \\
The function $f$ is the {\bf unknown}. \\
\parbox{11.8cm}{
\fbox{\parbox{5.8cm}{
$\bullet$ $f_t = f_{xx}$ heat equation \\
$\bullet$ $f_{tt} - f_{xx}=0$ wave equation \\
$\bullet$ $i \hbar f_{t} + f_{xx}=0$ Schr\"odinger 
}}
\fbox{\parbox{5.8cm}{
$\bullet$ $f_x-f_t=0$   transport equation \\
$\bullet$ $f_x f-f_t=0$ Burgers equation \\
$\bullet$ $f_{xx}+f_{yy}=0$ Laplace equation 
}}
}
}

\parbox{10cm}{
\fbox{II Facts}  \\
\parbox{11.8cm}{
\fbox{\parbox{5.8cm}{
$f_{xy} = f_{yx}$  {\bf Clairot's theorem}. \\
$\frac{d}{dt} f(\vec{r}(t)) = \nabla f(\vec{r}(t)) \cdot \vec{r}'(t)$. 
}}
\fbox{\parbox{5.8cm}{
$\vec{r}_u(u,v)$, $\vec{r}_v$ are 
tangent to a parametrized surface.
}}
\fbox{\parbox{11.8cm}{
the vector $\nabla f = \langle f_x,f_y,f_z \rangle$ is perpendicular
to the level surface $f(x,y,z) = c$. 
}}
Proof: differentiating 
$f(\vec{r}(t)) = c$ for a curve $\vec{r}(t)$ on the level set gives
$\frac{d}{dt} f(\vec{r}(t)) = 0$. The chain rule implies 
$\nabla f(\vec{r}(t)) \cdot \vec{r}'(t) = 0$ which shows that the
tangent vector to the curve is perpendicular to the level set. 
}
}

\parbox{10cm}{
\fbox{III Algorithms} \\
Distinguish ordinary and partial differential equations. 
Implicit differentiation: example: given $f(x,y(x))=0$, compute $y'(x)=-f_x/f_y$
without knowing $y(x)$: use chain rule $f_x 1 + f_y y'(x) = 0$. \\
An example in three dimensions: $f(x,y,z(x,y)) = 0$, 
compute $z_x = - f_x/f_z, z_y = -f_y/f_z$. Apply the chain rule to compute
derivatives. 
}

} \end{center} 





\begin{center} \parbox{12cm}{
Checklist Week 6:  \\

\parbox{11cm}{
\fbox{I Definitions}  \\
$\nabla f(x,y)=(f_x,f_y)$, $\nabla f(x,y,z) = (f_x,f_y,f_z)$, {\bf gradient} \\
$\nabla$ spelled "Nabla". 
$D_{\vec{v}} f = \nabla f \cdot \vec{v}$  {\bf directional derivative} (we define
it like this for all $\vec{v}$ !!!)
$L(\vec{x}) = f(\vec{x}_0) + \nabla f(\vec{x}_0) \cdot (\vec{x}-\vec{x}_0)$. 
linearization. \\
\fbox{\parbox{11.9cm}{
$L(x,y) = f(x_0,y_0) + f_x(x_0,y_0) (x-x_0) + f_y(x_0,y_0) (y-y_0)$ \\
{\bf linear approximation}.
}}
{\bf Tangent line}:  $L(x,y) = L(x_0,y_0)$ or \\
$a x + b y = d$ with $a=f_x(x_0,y_0),b=f_y(x_0,y_0)$, $d=a x_0 + b y_0$ \\
{\bf Tangent plane}:  $L(x,y,z) = L(x_0,y_0,z_0)$ or \\
Can {\bf estimate} $f(x,y,y)$ by $L(x,y,z)$ near $(x_0,y_0,z_0)$. \\
$f(x,y)$ {\bf differentiable} that is if $f_x,f_y$ continuous. \\

}

\parbox{11cm}{
\fbox{II Facts}  \\
$\frac{d}{dt}  f(\vec{r}(t)) = \nabla f(\vec{r}(t)) \cdot \vec{r} \; '(t)$ chain rule. \\
\fbox{\parbox{9.5cm}{$\nabla f(\vec{x}_0)$ orthogonal to level set
$f(\vec{x})=c$ containing$\vec{x}_0$.}}
$\frac{d}{dt} f(\vec{x}+t \vec{v}) = D_v f(\vec{x}$ by chain rule. \\
$(\vec{x}-\vec{x}_0)  \cdot \nabla f(\vec{x})=0$ {\bf tangent space} at $\vec{x}_0$ \\
Directional derivative maximal in $\vec{v}=\nabla f$ direction. \\
Partial derivatives are special directional derivatives. \\
If $D_vf (\vec{x}) = 0$ for all $\vec{v}$, then $\nabla f(\vec{x})=\vec{0}$. \\
}

\parbox{11cm}{
\fbox{III Algorithms} \\
Compute directional derivatives $D_{\vec{v}} f$.  \\
Find direction where directional derivative is maximal. \\
Find tangent lines and tangent planes.
Compute tangent lines and tangent planes. \\
Estimate $f(x_0+dx,y_0+dy)$ for small $dx$, $dy$. \\
Estimate $f(x,y)$ near a point $f(x_0,y_0)$ as $L(x,y)$, where
$L(x,y)$ is the linearization of $f$. 
}
} \end{center} 





\begin{center} \parbox{12cm}{
Checklist Week 7:  \\

\parbox{11cm}{
\fbox{I Definitions}  \\
$\nabla f(x,y)=(0,0)$, {\bf critical point} or {\bf stationary point}\\
$D=f_{xx} f_{yy} - f_{xy}^2$ {\bf discriminant}\\
$f(x_0,y_0) \geq f(x,y)$ in a neighborhood of $(x_0,y_0)$ {\bf local maximum} \\
$f(x_0,y_0) \leq f(x,y)$ in a neighborhood of $(x_0,y_0)$ {\bf local minimum} \\
$f(x_0,y_0) \geq f(x,y)$ for all $x,y$, global maximum or absolute maximum \\
$f(x_0,y_0) \leq f(x,y)$ for all $x,y$, global minimum or absolute minimum \\
$\nabla f(x,y) = \lambda \nabla g(x,y), g(x,y)=c$, $\lambda$ {\bf Lagrange multiplier}. \\
Two constraints: $\nabla f = \lambda \nabla g + \mu \nabla h, g=c,h=d$. \\
}

\parbox{11cm}{
\fbox{II Facts}  \\
Solutions to $\nabla f(x,y) = \lambda \nabla g(x,y), g(x,y)=c$ are candidates
for maxima or minima of $f$ under the constraint $g(x,y)=c$. 
\begin{center}
\fbox{\parbox{7cm}{
{\bf Second derivative test:} \\
$\nabla f=(0,0),D>0,f_{xx}<0$ local maximum \\
$\nabla f=(0,0),D>0,f_{xx}>0$ local minimum \\
$\nabla f=(0,0),D<0$  saddle point 
}}
\end{center}

\parbox{11cm}{
\fbox{III Algorithms} \\
Find and classify critical points in some domain. \\
Find maxima and minima for constrained optimization problems. \\
Find maxima and minima on a region with boundary (unconstrained problem
inside the region, Lagrange at the boundary). 
}
}

} \end{center} 





\begin{center} \parbox{12cm}{
Checklist Week 8:  \\

\parbox{11cm}{
\fbox{I Definitions}  \\

$\int \int_R f(x,y) \; dA$ {\bf double integral} \\
$\int_a^b \int_c^d f(x,y) \; dy dx$ integral over {\bf rectangle} \\
$\int_a^b \int_{c(x)}^{d(x)} f(x,y) \; dy dx$ {\bf y-simple, (type I) region} \\
$\int_c^d \int_{a(y)}^{b(y)} f(x,y) \; dx dy$ {\bf x-simple (type II) region} \\
}

\parbox{11cm}{
\fbox{II Facts}  \\

$\int_a^b \int_c^d f(x,y) \; dy dx = \int_c^d \int_a^b f(x,y) \; dx dy$ {\bf Fubini} \\
$\int \int_R 1 \; dx dy$   area of region $R$. \\
$\int \int_R f(x,y) \; dx dy = \int \int_U f(r \cos(\theta),r\sin(\theta)) \fbox{r} dr d\theta$ polar. \\
$\int \int_R f(x,y) \; dx dy$ volume of solid bounded above by graph(f) xy-plane.
}

\parbox{11cm}{
\fbox{III Algorithms} \\

Evaluate double integrals. \\
Calculate areas of regions. \\
Calculate volume of certain of bodies. \\
Switch order of integration. \\
Change from rectangular to polar coordinates. \\
Calculate areas of parameterized surfaces. 
} 

} \end{center} 





\begin{center} \parbox{12cm}{
Checklist Week 9:  \\

\parbox{11cm}{
\fbox{I Definitions}  \\

$\int \int_R  |\vec{r}_u \times \vec{r}_v| \; du dv$ {\bf surface area} \\
$\int \int \int_R f(x,y,z) \; dV$ {\bf triple integral} \\
$\int_a^b \int_c^d \int_u^v f(x,y,z) \; dy dx$ integral over {\bf rectangular box} \\
$\int \int \int_R  f(r,\theta,z)$ \fbox{$r$} $\; dz dr d\theta$ {\bf cylindrical coordinates} \\
$\int \int \int_R  f(\rho,\theta,z)$ \fbox{$\rho^2 \sin(\phi)$} $\; dz dr d\theta$
                   {\bf spherical coordinates} \\
} \\

\parbox{11cm}{
\fbox{II Facts}  \\

$V=\int \int \int_R 1 \; dV$ {\bf volume} of solid $R$. \\
$M=\int \int \int_R \rho(x,y,z) \; dV$  {\bf mass} of solid $R$ with density $\rho$. \\
$(\int\int\int_R x \; dV/V,\int\int\int_R y \; dV/V,\int\int\int_R z \; dV/V)$ {\bf center} of mass. \\
}  \\

\parbox{11cm}{
\fbox{III Algorithms} \\

Switch order of integration in triple integrals.
Apply the right coordinate system for integration.
Make a picture of the region one has to integrate over.
}

\parbox{11cm}{
\fbox{II Facts}  \\

$\int_a^b \int_c^d f(x,y ) \; dx dy =  \int_c^d \int_a^b f(x,y) \; dx dy$ {\bf Fubini} \\
Formulas for surface area in case of sphere, surface of revolution and graph. \\
}

\parbox{11cm}{
\fbox{III Algorithms} \\
Change order of integration. \\
Use correct coordinate system. \\
Get from integral the region and from the region the integral. \\
} 
} \end{center}






\begin{center} \parbox{12cm}{
Checklist Week 10:  \\

\parbox{11cm}{
\fbox{I Definitions}  \\
$\vec{F}(x,y) = \langle P(x,y),Q(x,y) \rangle$ vector field in 2D, \\
$\vec{F}(x,y,z) = \langle P(x,y,z),Q(x,y,z),R(x,y,z) \rangle$ vector field in 3D, \\
$\int_{\gamma} F \cdot dr = \int_a^b F(r(t)) \cdot r'(t) \; dt$ line integral
$\vec{F}(x,y) = \nabla f(x,y)$  gradient field = potential field = conservative,
$f$ is called the {\bf potential} of $F$. \\
} \\

\parbox{11cm}{
\fbox{II Facts}  \\

Line integrals are independent of the parametrization. \\
Line integrals are positive if the angle between the velocity vector and the 
field is acute at all points of the curve. \\
Line integrals have the interpretation of "Work", if $F$ is interpreted as
a force field. 
If $Q_x-P_y \neq 0$ at some place, then $F=(P,Q)$ is not a gradient field
$F=(f_x,f_y)$.  (Clairot).
}  \\

\parbox{11cm}{
\fbox{III Algorithms} \\

Match vector fields with formulas.
Compute line integrals for a given parametrized curve and a given vector field. 
See, whether a field is not a gradient field, by checking with Clairot.

} 

} \end{center} 





\begin{center} \parbox{12cm}{
Checklist Week 11:  \\

\parbox{14cm}{
\fbox{I Definitions}  \\
$F(x,y)=(P,Q)$, ${\rm curl}(F) = Q_x-P_y$. ${\rm div}(F)=P_x+Q_y$. \\
$F(x,y,z)=(P,Q,R)$, ${\rm curl}(P,Q,R)=(R_y-Q_z,P_z-R_x,Q_x-P_y)$, ${\rm div}(P,Q,R)=P_x+Q_y+R_z$. \\
$\Delta f = {\rm div} {\rm grad}(f) = f_{xx} + f_{yy} + f_{zz}$ Laplacian for functions \\

\parbox{10cm}{
\parbox{2cm}{NABLA calculus:}
\fbox{\parbox{3.5cm}{
$\nabla = (\partial_x, \partial_y, \partial_z)$, ${\rm grad}(f) = \nabla f$,
${\rm curl}(\vec{F}) = \nabla \times \vec{F}$, ${\rm div}(f) = \nabla \cdot \vec{F}$.  
}}
\fbox{\parbox{3.5cm}{
${\rm div}(F) = \nabla \cdot F$ \\
$\Delta f = \nabla \cdot \nabla f$ \\
}}
}
}

\parbox{14cm}{
\fbox{II Facts}  \\

Fundamental theorem of line integrals: $F(x,y) = \nabla f(x,y)$
\begin{center}
\fbox{$\int_a^b F(r(t)) \cdot r'(t)) \; dt = f(r(b)) - f(r(a))$}
\end{center}
$F$ defined in in a simply connected region $R$
$\int_{\gamma} F \; dr = 0$, for every closed curve $\gamma$. \\
{\bf Green's theorem:} $C$ boundary of $R$, ($R$ is "to the left")
\begin{center}
\fbox{ \parbox{6cm}{
$\int_{C} F \cdot dr = \int \int_R {\rm curl}(F) \; dx dy$
}} 
\end{center} 

\fbox{ \parbox{7.5cm}{
${\rm div}({\rm curl}(F)) = 0$. \\
${\rm curl}({\rm grad}(F)) = \vec{0}$ \\
${\rm curl}({\rm curl}(F)) = {\rm grad}({\rm div}(F)) - \Delta(F)$.
}}

}  \\

\parbox{14cm}{
\fbox{III Algorithms} \\
Find curl and divergence for specific vector fields $F$.
Decide whether a vector field $F$ can be of the form $F={\rm curl}(G)$.
Decide whether a vector field $F$ can be of the form $F={\rm grad}(f)$. 
Use FTL to switch between line integrals and summation of end points. 
Use Greens theorem to switch between double and line integrals. 
Calculate with $\nabla$ (Nabla calculus). 
} 

} \end{center} 





\begin{center} \parbox{12cm}{
Checklist Week 12:  \\

\parbox{11cm}{
\fbox{I Definitions}  \\

$F(x,y,z)$ vector field, $S=r(R)$ parametrized surface \\
$r_u \times r_v$ normal vector, $\vec{n}=\frac{r_u \times r_v}{|r_u \times r_v|}$ unit normal vector  \\
$|r_u \times r_v| du dv = dS$ surface element \\
$r_u \times r_v du dv = \vec{dS}=\vec{n} dS$ normal surface element. \\
$\int\int_S f \; dS = \int\int_S f(r(u,v)) |r_u \times r_v| \; du dv$ {\bf surface integral} \\
$\int\int_S F \; \vec{dS} = \int\int_S F(r(u,v)) \cdot (r_u \times r_v) \; du dv$ {\bf flux integral}. \\
}

\parbox{11cm}{
\fbox{II Facts}  \\

{\bf Stokes's theorem:} $C$ boundary of surface $S$
\begin{center}
\fbox{ \parbox{6cm}{
$\int_{C} F \cdot dr = \int \int_S {\rm curl}(F) \cdot dS$
}} 
\end{center} 

{\bf Gauss (Divergence) theorem:} $S$ boundary of region $E$
\begin{center}
\fbox{ \parbox{6cm}{
$\int \int_{S} F \cdot dS = \int \int \int_E {\rm div}(F) \; dV$
}}
\end{center}


}  \\

\parbox{11cm}{
\fbox{III Algorithms} \\

Use Stokes theorem to switch between flux and line integrals. \\
Use Gauss theorem to switch between triple and flux integrals. \\
} 

} \end{center} 


\end{document}
