\contentsline {part}{I\hspace {1em}Real Multivariate and vector functions}{1}{}%
\contentsline {chapter}{\numberline {1}Multivariate functions}{3}{}%
\contentsline {chapter}{\numberline {2}Multivariate limits}{5}{}%
\contentsline {chapter}{\numberline {3}Multivariate continuity}{7}{}%
\contentsline {chapter}{\numberline {4}Multivariate differentiation}{9}{}%
\contentsline {section}{\numberline {4.1}Directional derivative}{9}{}%
\contentsline {section}{\numberline {4.2}Partial derivative}{10}{}%
\contentsline {section}{\numberline {4.3}The Jacobian matrix and Hessian matrix}{11}{}%
\contentsline {chapter}{\numberline {5}Total derivative}{13}{}%
\contentsline {subsection}{\numberline {5.0.1}Properties of Jacobian matrix}{15}{}%
\contentsline {section}{\numberline {5.1}Multivariate chain rule}{16}{}%
\contentsline {section}{\numberline {5.2}Clairaut's theorem}{16}{}%
\contentsline {section}{\numberline {5.3}Multivariate inverse function theorem}{16}{}%
\contentsline {subsection}{\numberline {5.3.1}Multivariate inverse function theorem}{17}{}%
\contentsline {subsection}{\numberline {5.3.2}Multivariate implicit function theorem}{17}{}%
\contentsline {section}{\numberline {5.4}Multivariate Taylor's theorem}{17}{}%
\contentsline {subsection}{\numberline {5.4.1}Multivariate Taylor's theorem}{17}{}%
\contentsline {subsection}{\numberline {5.4.2}Multivariate Taylor series}{17}{}%
\contentsline {section}{\numberline {5.5}Derivative tests}{17}{}%
\contentsline {subsection}{\numberline {5.5.1}Properties of the Hessian matrix}{17}{}%
\contentsline {subsection}{\numberline {5.5.2}Lagrangian multipliers}{17}{}%
\contentsline {chapter}{\numberline {6}Multivariate integration}{19}{}%
\contentsline {section}{\numberline {6.1}Multivariate integration}{19}{}%
\contentsline {section}{\numberline {6.2}Fubini's theorem and Tonelli's theorem}{19}{}%
\contentsline {subsection}{\numberline {6.2.1}Nested integration}{19}{}%
\contentsline {subsection}{\numberline {6.2.2}Fubini's theorem}{19}{}%
\contentsline {subsection}{\numberline {6.2.3}Tonelli's theorem}{19}{}%
\contentsline {section}{\numberline {6.3}Fubini's theorem and Tonelli's theorem}{20}{}%
\contentsline {part}{II\hspace {1em}Differentiable curves and surfaces}{25}{}%
\contentsline {chapter}{\numberline {7}Differentiable curves}{27}{}%
\contentsline {section}{\numberline {7.1}Differentiable curve}{27}{}%
\contentsline {section}{\numberline {7.2}Closed differentiable curve}{28}{}%
\contentsline {section}{\numberline {7.3}Line integral (scalar field)}{28}{}%
\contentsline {chapter}{\numberline {8}Differentiable surfaces}{31}{}%
\contentsline {section}{\numberline {8.1}Differentiable surface}{31}{}%
\contentsline {section}{\numberline {8.2}Various parametrization}{31}{}%
\contentsline {section}{\numberline {8.3}Surface integral}{31}{}%
\contentsline {section}{\numberline {8.4}Flux integral}{31}{}%
\contentsline {part}{III\hspace {1em}Frames}{33}{}%
\contentsline {chapter}{\numberline {9}Frames of coordinate systems}{35}{}%
\contentsline {section}{\numberline {9.1}Applications for line integrals}{36}{}%
\contentsline {section}{\numberline {9.2}Applications for surface and flux integrals}{36}{}%
\contentsline {chapter}{\numberline {10}Frenet-Serret frame}{37}{}%
\contentsline {part}{IV\hspace {1em}Multivariate and vector differential operators}{39}{}%
\contentsline {chapter}{\numberline {11}Nabla symbol}{41}{}%
\contentsline {chapter}{\numberline {12}Multivariate differential operators}{43}{}%
\contentsline {section}{\numberline {12.1}Gradient operator}{43}{}%
\contentsline {section}{\numberline {12.2}Laplace operator}{44}{}%
\contentsline {chapter}{\numberline {13}Vector differential operators}{45}{}%
\contentsline {section}{\numberline {13.1}Divergence operator}{45}{}%
\contentsline {section}{\numberline {13.2}Curl operator}{45}{}%
\contentsline {chapter}{\numberline {14}Differential operators in different frames}{47}{}%
\contentsline {chapter}{\numberline {15}Integral representation}{49}{}%
\contentsline {chapter}{\numberline {16}Introduction to the integral theorems}{51}{}%
\contentsline {chapter}{\numberline {17}That's all folks}{55}{}%