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\contentsline {section}{\numberline {1}Coordinate Systems and Polar Functions}{4}{section.1}%
\contentsline {subsection}{\numberline {1.1}Rectangular Coordinates}{4}{subsection.1.1}%
\contentsline {subsection}{\numberline {1.2}Polar Coordinates}{4}{subsection.1.2}%
\contentsline {subsection}{\numberline {1.3}Transformations}{4}{subsection.1.3}%
\contentsline {subsection}{\numberline {1.4}Polar Graphs}{5}{subsection.1.4}%
\contentsline {subsection}{\numberline {1.5}Differentiating Polar Functions}{5}{subsection.1.5}%
\contentsline {subsubsection}{\numberline {1.5.1}Chain Rule}{6}{subsubsection.1.5.1}%
\contentsline {subsubsection}{\numberline {1.5.2}Polar Derivatives}{6}{subsubsection.1.5.2}%
\contentsline {subsection}{\numberline {1.6}Arc Lengths}{6}{subsection.1.6}%
\contentsline {subsubsection}{\numberline {1.6.1}Review}{6}{subsubsection.1.6.1}%
\contentsline {subsubsection}{\numberline {1.6.2}Polar Arc Length}{7}{subsubsection.1.6.2}%
\contentsline {subsection}{\numberline {1.7}Polar Area}{7}{subsection.1.7}%
\contentsline {section}{\numberline {2}Vectors}{7}{section.2}%
\contentsline {subsection}{\numberline {2.1}Fundamentals}{7}{subsection.2.1}%
\contentsline {subsection}{\numberline {2.2}Norms}{8}{subsection.2.2}%
\contentsline {subsection}{\numberline {2.3}Dot Product}{9}{subsection.2.3}%
\contentsline {subsubsection}{\numberline {2.3.1}Angles in the Dot Product}{9}{subsubsection.2.3.1}%
\contentsline {subsubsection}{\numberline {2.3.2}Projections}{9}{subsubsection.2.3.2}%
\contentsline {subsection}{\numberline {2.4}Three Dimensions}{10}{subsection.2.4}%
\contentsline {subsubsection}{\numberline {2.4.1}Set Notation}{10}{subsubsection.2.4.1}%
\contentsline {subsection}{\numberline {2.5}Cross Product}{10}{subsection.2.5}%
\contentsline {subsubsection}{\numberline {2.5.1}Properties}{11}{subsubsection.2.5.1}%
\contentsline {subsection}{\numberline {2.6}Matrices}{11}{subsection.2.6}%
\contentsline {subsubsection}{\numberline {2.6.1}2 by 2}{11}{subsubsection.2.6.1}%
\contentsline {subsubsection}{\numberline {2.6.2}3 by 3}{11}{subsubsection.2.6.2}%
\contentsline {subsubsection}{\numberline {2.6.3}Calculating the Cross Product With Matrices}{12}{subsubsection.2.6.3}%
\contentsline {subsection}{\numberline {2.7}Parametric Equations}{12}{subsection.2.7}%
\contentsline {subsubsection}{\numberline {2.7.1}Some Notation}{12}{subsubsection.2.7.1}%
\contentsline {subsubsection}{\numberline {2.7.2}Crossing Sets}{12}{subsubsection.2.7.2}%
\contentsline {subsection}{\numberline {2.8}Lines in 3 Dimensions}{13}{subsection.2.8}%
\contentsline {subsection}{\numberline {2.9}Distance Between Line and Point}{13}{subsection.2.9}%
\contentsline {subsection}{\numberline {2.10}Planes}{14}{subsection.2.10}%
\contentsline {subsubsection}{\numberline {2.10.1}Given Three Points}{14}{subsubsection.2.10.1}%
\contentsline {subsubsection}{\numberline {2.10.2}Finding line Given Two Planes}{14}{subsubsection.2.10.2}%
\contentsline {subsubsection}{\numberline {2.10.3}Distance From Point To Plane}{14}{subsubsection.2.10.3}%
\contentsline {section}{\numberline {3}Parametric Equations}{15}{section.3}%
\contentsline {subsection}{\numberline {3.1}Example}{15}{subsection.3.1}%
\contentsline {subsection}{\numberline {3.2}Derivatives of Parametric Equations}{15}{subsection.3.2}%
\contentsline {subsection}{\numberline {3.3}Arclength}{15}{subsection.3.3}%
\contentsline {subsection}{\numberline {3.4}Constant Length}{15}{subsection.3.4}%
\contentsline {subsection}{\numberline {3.5}Special Unit Vectors}{16}{subsection.3.5}%
\contentsline {subsection}{\numberline {3.6}Curvature}{16}{subsection.3.6}%
\contentsline {subsection}{\numberline {3.7}Decomposing Acceleration}{17}{subsection.3.7}%
\contentsline {subsubsection}{\numberline {3.7.1}Derivation}{17}{subsubsection.3.7.1}%
\contentsline {subsection}{\numberline {3.8}Integration}{17}{subsection.3.8}%
\contentsline {subsection}{\numberline {3.9}Projectile Motion}{18}{subsection.3.9}%
\contentsline {section}{\numberline {4}Functions of Several Variables}{18}{section.4}%
\contentsline {subsection}{\numberline {4.1}Limits}{18}{subsection.4.1}%
\contentsline {subsubsection}{\numberline {4.1.1}Delta Epsilon Definition of The Limit}{19}{subsubsection.4.1.1}%
\contentsline {subsection}{\numberline {4.2}Partial Derivatives}{19}{subsection.4.2}%
\contentsline {subsubsection}{\numberline {4.2.1}Linearity}{19}{subsubsection.4.2.1}%
\contentsline {subsubsection}{\numberline {4.2.2}Notation}{20}{subsubsection.4.2.2}%
\contentsline {subsection}{\numberline {4.3}Chain Rule}{20}{subsection.4.3}%
\contentsline {subsection}{\numberline {4.4}Level Curves}{21}{subsection.4.4}%
\contentsline {subsection}{\numberline {4.5}Directional Derivatives}{21}{subsection.4.5}%
\contentsline {subsection}{\numberline {4.6}Matrices}{21}{subsection.4.6}%
\contentsline {subsection}{\numberline {4.7}Hessian}{22}{subsection.4.7}%
\contentsline {subsection}{\numberline {4.8}Taylor Expansions}{22}{subsection.4.8}%
\contentsline {subsubsection}{\numberline {4.8.1}Example}{22}{subsubsection.4.8.1}%
\contentsline {subsubsection}{\numberline {4.8.2}Taylor In 2 Variables}{22}{subsubsection.4.8.2}%
\contentsline {subsection}{\numberline {4.9}Tangent Planes}{23}{subsection.4.9}%
\contentsline {subsubsection}{\numberline {4.9.1}Level Surfaces}{23}{subsubsection.4.9.1}%
\contentsline {section}{\numberline {5}Optimization}{23}{section.5}%
\contentsline {subsection}{\numberline {5.1}Unconstrained optimization}{23}{subsection.5.1}%
\contentsline {subsubsection}{\numberline {5.1.1}Classification}{24}{subsubsection.5.1.1}%
\contentsline {subsection}{\numberline {5.2}Constrained Optimization}{24}{subsection.5.2}%
\contentsline {subsubsection}{\numberline {5.2.1}Example Problem}{25}{subsubsection.5.2.1}%
\contentsline {subsection}{\numberline {5.3}Lagrange Multipliers}{26}{subsection.5.3}%
\contentsline {section}{\numberline {6}Iterated Integrals}{26}{section.6}%
\contentsline {subsection}{\numberline {6.1}Complex Regions}{26}{subsection.6.1}%
\contentsline {subsection}{\numberline {6.2}Calculating Areas}{27}{subsection.6.2}%
\contentsline {subsection}{\numberline {6.3}Splitting Regions}{27}{subsection.6.3}%
\contentsline {subsection}{\numberline {6.4}Polar Regions}{27}{subsection.6.4}%
\contentsline {subsubsection}{\numberline {6.4.1}Jacobians}{28}{subsubsection.6.4.1}%
\contentsline {subsection}{\numberline {6.5}Triple Integrals}{28}{subsection.6.5}%
\contentsline {subsection}{\numberline {6.6}Applications}{28}{subsection.6.6}%
\contentsline {subsubsection}{\numberline {6.6.1}Center of Mass}{28}{subsubsection.6.6.1}%
\contentsline {subsection}{\numberline {6.7}Angular Motion}{29}{subsection.6.7}%
\contentsline {subsection}{\numberline {6.8}Three Dimensional Coordinate Systems}{29}{subsection.6.8}%
\contentsline {subsubsection}{\numberline {6.8.1}Rectangular Coordinates}{29}{subsubsection.6.8.1}%
\contentsline {subsubsection}{\numberline {6.8.2}Cylindrical Coordinates}{29}{subsubsection.6.8.2}%
\contentsline {subsubsection}{\numberline {6.8.3}Spherical Coordinates}{30}{subsubsection.6.8.3}%
\contentsline {subsection}{\numberline {6.9}Line Integrals}{30}{subsection.6.9}%
\contentsline {subsection}{\numberline {6.10}General Change of Variable}{31}{subsection.6.10}%