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\markright{Math 208 Exam Topics Review Summer 2006\hfill Page }

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\centerline{\textbf{Exam 1 Topics Review}}

\begin{enumerate}
\item Vectors in ${\Bbb R}^2$
\begin{enumerate}
\item Vectors and scalars. The magnitude and direction of a vector.
\item Graphical vector operations: addition, subtraction, scalar
multiplication.
\item Vectors in component form. Vectors in polar form.
\item Vector algebra with components: addition, subtraction, scalar
multiplication.
\item The zero vector. The standard basis vectors $\vec i,\vec j$.
Vector as a linear combination of the standard basis vectors.
\end{enumerate}

\item Vectors in ${\Bbb R}^3$
\begin{enumerate}
\item Graphical vector operations: addition, subtraction, scalar
multiplication.
\item The standard basis vectors $\vec i,\vec j, \vec k$.
Vector as a linear combination of the standard basis vectors.
\item Magnitude, length of a vector.
\item Parallel vectors. Unit vectors.
\item Vector algebra with components: addition, subtraction, scalar
multiplication.
\end{enumerate}


\item The dot product of vectors.
\begin{enumerate}
\item Definition of the dot product in component form.
\item The cosine formula for the dot product.
\item Basic properties of the dot product in terms of algebraic operations.
\item Angle between two vectors. Orthogonal vectors.
\item Dot production condition for orthogonal vectors.
\item Component of a vector in another. Projection of a vector in
another.
\end{enumerate}

\item The cross product of vectors in ${\Bbb R}^3$.
\begin{enumerate}
\item $2\times 2, 3\times 3$ determinants.
\item Definition of the cross product in component form.
\item The right-hand rule for the direction of the cross product.
\item Magnitude of the cross product. The area of a parallelogram,
triangle. The volume of a parallelpiped.
\item Basic properties of the cross product in terms of algebraic operations.
\end{enumerate}

\item Lines ${\Bbb R}^3$.
\begin{enumerate}
\item Direction of a line. Parametric equations of lines.
Symmetrical equations of lines.
\item Equation of a line through two points.
\item Find intersection of two lines.
\item Distance from a point to a line. Distance between two lines.
\end{enumerate}

\item Planes ${\Bbb R}^3$.
\begin{enumerate}
\item Normal direction of of a plane.
\item Equation of a plane through a point with a normal vector.
\item Equation of a plane through 3 points.
\item Determine normal vectors for a plane using cross product.
\item Condition of two parallel planes.
\item Distance from a point to a plane. Distance between a line and
a plane. Distance between 2 planes.
\end{enumerate}

\item Vector-valued functions.
\begin{enumerate}
\item Functions from $\Bbb R$ to ${\Bbb R}^2$ or ${\Bbb R}^3$.
\item Equations of curves. Parameterized equations of curves.
Orientation of a curve by its parameterization.
\item Parameterizations of simple curves: lines, circles, ellipses,
parabolas, helixes.
\item Calculus of vector-valued functions: differentiation and
integration.
\item Tangent vector of a curve.
\item Arc length of a curve.
\end{enumerate}

\item Motion in space.
\begin{enumerate}
\item Position vector, velocity vector, and acceleration vector.
\item Unite tangent vector. The principal unit normal vector.
\item Arc length parameterization. Speed. Curvature of a curve.
\item Tangential component of acceleration, normal component of
acceleration.
\item Acceleration vector decomposed in its tangential and principal
normal components.
\item Newton's second law of motion. Newton's law of gravity.
Projectile motion.
\end{enumerate}

\item Function of several variables.
\begin{enumerate}
\item Real valued functions taking ${\Bbb R}^2$ or ${\Bbb R}^3$ to
$\Bbb R$. Domain and range of a function.
\item Graph of a 2 variable function. Level curves, contours,
trances of a function of 2 variables.
\item Level surfaces of functions of 3 variables.
\end{enumerate}

\item Limits of functions of several variables.
\begin{enumerate}
\item Evaluation of limits with singularities.
\item Non-existence of limit by 2-path rule.
\item Continuity of functions.
\end{enumerate}

\end{enumerate}





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