# Calculus in Several Variable

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## Overview

### Module 1. 3-dimensional geometry and functions of several variables

#### Lecture 1

• 10.1 Analytic geometry in three dimensions

Exercises: 11, 25, 27, 29, 31, 33, 35, 37, 39

• 10.6 Cylindrical and spherical coordinates

Exercises: 3, 5, 9, 13

#### Lecture 2

• 11.1 Vector values functions in on variable

Exercises: 17, 21, 33

• 11.2 Applications of vector differentiation

Exercises: 3

• 11.3 Curves and parametrizations

Exercises: 5, 7, 11, 13, 15

#### Lecture 3

• 12.1 Functions in several variables

Exercises: 5, 9, 13,15, 17, 23, 27, 33

• 12.2 Limits and continuity

Exercises: 5, 7, 9, 11, 15

### Module 2. Partial derivatives and linear approximation

#### Lecture 4

• 12.3 Partial derivatives

Exercises: 5, 7, 13, 23

• 12.4 Higher order derivatives

Exercises: 5, 7, 11, 15, 17

• 12.5 The chain rule

Exercises: 7, 11, 17, 21

#### Lecture 5

• 12.6 Linear approximation, differentiability and differentials

Exercises: 3, 5, 17, 19

• 12.7 Gradient and directional derivatives

Exercises: 3, 5, 13, 17, 25

### Module 3. Applications of derivatives

#### Lecture 6

• 12.8 Implicit functions

Exercises: 13, 17

• 12.9 Taylor's formula, Taylor series and approximation

Exercises: 1, 3, 5, 7, 11

#### Lecture 7

• 13.1 Extreme values

Exercises: 5, 7, 9, 19, 23, 25

• 13.2 Extreme values of functions with constraints

Exercises: 3, 5, 9, 15

#### �Lecture 8

• 13.3 Lagrange multipliers

Exercises: 3, 9, 11, 15

• 13.4 Lagrange multipliers in higher-dimensional spaces

Exercises: 1, 3

### Module 4. Multiple integrals

#### Lecture 9

• 14.1 Double integrals

Exercises: 15, 19, 21

• 14.2 Iterated integration in cartesian coordinates

Exercises: 3, 5, 15, 23

#### Lecture 10

• 14.3 Generalized integrals and the mean value theorem

Exercises: 1, 3, 13, 27

• 14.4 Double integrals in polar coordinates

Exercises: 5, 9, 15, 19, 21

#### Lecture 11

• 14.5 Tripple integrals

Exercises: 5, 7, 9

• 14.6 Change of variables in triple integrals

Exercises: 3, 7, 11

• 14.7 Applications of multiple integrals

Exercises: 5, 9, 13, 21,27

### Module 5. Line and surface integrals

#### Lecture 12

• 15.1 Vector fields and scalar fields

Exercises: 3, 5, 17

• 15.2 Conservative fields

Exercises: 3, 5, 7, 21

#### Lecture 13

• 15.3 Line integrals

Exercises: 7, 11

• 15.4 Line integrals of vector fields

Exercises: 1, 5, 7, 15

#### Lecture 14

• 15.5 Surfaces and surface integrals

Exercises: 1, 7, 13

• 15.6 Oriented integrals and flux integrals

Exercises: 5, 9, 13, 15

### Module 6. Vector Calculus

#### Lecture 15

• 16.1 Gradient, divergence and curl

Exercises: 3, 7, 11

• 16.2 Some identities involving grad, div and curl

Exercises: 9, 15, 17

#### Lecture 16

• 16.3 Green's Theorem in the plane

Exercises: 3, 5, 9

#### Lecture 17

• 16.4 The Divergence Theorem in three-dimensional space

Exercises: 5, 11, 15

• 16.5 Stoke's Theorem in three-dimensional space

Exercises: 1, 3, 5

#### Lecture 18

• Recap for examination