The offered compendium of lectures is considered to help foreign students training profession 6.050304 "Oil and Gas Engineering" to learn and study how to solve problems from the course of differential and integral calculation of functions of several variables. The theory is given with figures, sums samples. Конспект лекцій розроблено відповідно до робочої програми дисципліни "Вища математика". Призначено для студентів стаціонарної форми навчання за напрямом підготовки 6.050304 - "Нафтогазова інженерія". CONTEXT Differentiation of Several Variables Function Lecture 1 Notion of Several Variables Function, Its Limit, Continuity and Partial Derivative................................................ 7 1.1 Determination of Several Variables Function.............. 7 1.2 Lines and Level Surfaces...............................................9 1.3 Limit of Several Variables Function........................... 10 1.4 Continuity of Several Variables Function.................. 14 1.5 Partial and Full Change of Height for Several Variables Functions..................................................................... 16 1.6 Partial Differentiations for Several Variables Functions and Their Geometrical Maintenance....................... 18 Lecture 2 Differentiation of Several Variables Function ...... 24 2.1 Definition of several variables function differentiation, connection with continuity and existence of partial derivatives......................................................24 2.2 Differential of Several Variables Function.................. 28 2.3 Partial Derivatives and Differential of a Complete Function.......................................................................................32 2.4 Differentiation of Non-Obvious Functions.................. 37 Lecture 3 Some Applications of Partial Derivatives............... 44 3.1 The Tangent Plane and the Normal to the Surface. The Geometrical Maintenance of the Differential of Two Variables Function........ 44 3.2 Scalar Field. Derivative due to Direction.................... 48 Lecture 4 Partial Derivatives and Differentials of Higher Orders. Taylor's Formula........................................................... 58 4.1 Partial Derivatives of Higher Orders.......................... 58 4.2 Differentials of Higher Orders......................................64 4.3 Taylor's Formula for the Function of Two Variables. 71 Lecture 5 Extremes of Several Variables Functions............... 78 5.1 Local Extremes of Several Variables Functions. Necessary Conditions of Extreme....................................... 78 5.2 Sufficient Conditions of Extreme................................81 5.3 Global Extremes of Several Variables Functions....... 87 5.4 Conditional Extreme..................................................91 Integration of Several Variables Function Lecture 1 Double integral. Principal concepts and definition. Conditions of the existence and properties of calculation. Change of variable in double integral. Application of double integrals………………………………………………………… 98 1.1 Problem Leading to the Concept of Double Integral.. 98 1.1.1 Problem of Computing of Mass of a Material Plate… 98 1.1.2 Problem of Computing of Volume of a Cylindrical Body…………………………………………………………….. 99 1.2 Definition and the Condition of the Existence of a Double Integral……………………………………………….. 101 1.3. Principal Properties………………………………… 103 1.4 Reducing a Double Integral to an Iterated Integral.. 104 1.4.1 The Case of a Rectangular Domain……………. 104 1.5 Changing of Variables in Double Integral…………. 113 1.5.1 Cartesian Coordinates ...………………………. 113 1.5.2 Integral in Polar Coordinates………………..… 116 1. 6 Application of Double Integrals………………….... 118 1.6.1 Volume of cylindrical body…………………….. 118 1.6.2 Area of plane figures…………………………… 119 1.6.3 Surface area…………………………………….. 119 1.6.4 Mass of a planar figure………………………… 119 1.6.5 Coordinates of the center of mass of a plate….. 120 1.6.6 Static moment of inertia at a planar figure relative to coordinates axis……………………………………………. 120 1.6.7 Moment of inertia………………………………. 120 Lecture 2 Triple integrals. Principal concepts and definitions. Conditions of the existence and properties. Cylindrical and Spherical coordinates. Change of variable in double integral. Application of double integrals……………………………… 130 2.1 Principal Definitions. Condition for the Existence of a Triple Integral………………………………………………… 130 2.2 Properties of Triple Integrals……………………….. 132 2.3 Triple Integral in Cartesian Coordinates………….. 133 2.3 Triple Integral in Cylindrical and Spherical Coordinates……136 2.3.1 Cartesian Coordinates………………………….. 136 2.3.2 Triple Integral in Cylindrical Coordinates….... 138 2.3.3 Triple Integral in Spherical Coordinates……… 141 2.4. Application of Triple Integrals…………………….... 145 Lecture 3 Line Integrals Basic Concepts……………………. 152 3.1 Line Integrals of the First Type. Basic Concepts…… 152 3.2 Geometrical Significance of a Line Integral of the First Type……………………………………………………………. 154 3.3 Evaluation of a Line Integral of the First Type…….. 154 3.4 Line Integrals of the Second Type. Basic Concepts... 159 3.5 Evaluation and Properties of Line Integrals of the Second Type…………………………………………………... 161 3.5 Green Formula……………………………………….. 166 3.6 Independence of a Line Integral of a Path of Integration…………………………………………………….. 168 3.7 Application of Line Integral………………………… 172 Lecture 4 Surface integrals of the first and of the second type. Properties and evaluation. Ostrograsky-Gauss formula. Stokes formula……………………………………179 4.1 Surface Integrals of the First Type. Basic Concepts. 179 4.2 Evaluation of the Surface Integral of the First Type……………………………………………………… 180 4.3 Surface Integrals of the Second Type. The Basic Concepts……………………………………………………… 182 4.4 Evaluation of the Surface Integral of the Second Type………………………………………………………. 185 4.5 Ostrogradsky - Gauss Formula…………………… 187 4.6 Stokes Formula………………………………………. 189 4.7 Applications of Surface Integrals………………….. 190 Lecture 5 Scalar and vector fields. Gradient of a scalar field. Directional derivative. Flux, circulation, divergence, rotation of the vector field. Gauss - Ostrogradsky formula. Stokes formula. Hamiltonian operator. Potential, solenoid, harmonic fields. Differential operations of the first and second orders………………………………………………………….. 203 5.1 Basic Concepts of Field Theory…………………...… 203 5.2 Scalar Field………………………………………….... 204 5.2.1. Directional Derivative………………………….. 204 5.2.2 Gradient of the Scalar Field and its Properties. 207 5.3 Vector Field…………………………………………... 209 5.3.1 Vector Lines……………………………………... 209 5.3.2 Flux of a Vector Across a Surface……………... 210 5.3.3 Field Divergence. Gauss - Ostrogradsky Formula in a Vector Form……………………………………………… 213 5.3.4 Circulation of a Vector Field. Rotation of a Vector. Stokes Formula in a Vector Form………………………. 215 5.3.5 Hamiltonian Operator. Differential Operations of the First and Second Orders………………………………… 217 5.3.6 Properties of Vector Fields. Solenoid Field. Properties of the solenoid field………………………….. 220 References..............................................................................228