Mathematical Analysis 2 Lecture 1 - Functions of two or more independent variables. Differential. Partial derivatives with constrained variables Aigul Myrzagaliyeva aigul.myrzagalieva@astanait.edu.kz Astana IT University Lecture overview • Multivariable functions • Interior and boundary points, open and closed region • Level Curve, Graph of function • Limits of multivariable functions • Continuity of multivariable functions • Partial differentiation • Chain rules for partial differentiation Multivariable functions Multivariable functions Interior and boundary points, open and closed region Interior and boundary points, open and closed region Interior and boundary points, open and closed region Interior and boundary points, open and closed region Interior and boundary points,open and closed region Level Curve, Graph of function Level Curve, Graph of function Level surface, Limits of multivariable functions Limits along smooth curves Limits of multivariable functions Limits of multivariable functions Limits along smooth curves Continuity of multivariable functions Theorem: Continuity of multivariable functions Continuity of multivariable functions Partial differentiation Partial differentiation Partial differentiation Partial differentiation Partial differentiation Partial differentiation Partial differentiation Partial differentiation Partial differentiation f x, y x cos y ye x df cos y ye x dx d df d cos y ye x dy dx dy d2 f sin y e x dy dx f y x sin y e x f x, y x cos y ye x df x sin y e x dy d df d x sin y e x dx dy dx d2 f sin y e x dx dy f xy sin y e x Partial differentiation Partial differentiation Chain rules for partial differentiation Definition (chain rules): Chain rules for partial differentiation Chain rules for partial differentiation Definition (chain rules) for partial derivatives: Chain rules for partial differentiation Chain rules for partial differentiation Chain rules for partial differentiation Chain rules for partial differentiation Lecture Summary • Multivariable functions • Limits of multivariable functions • Limits along smooth curves • Continuity of multivariable functions • Partial differentiation • Chain rules for partial differentiation
