Calculus II

Course Page

Course Information

• Calculus II (UoG11014.06)
• This is an English taught course for first-year undergraduate students.
• This course extends the basic operation skills of calculus to infinite series, differential calculus for multivariable functions, integral calculus for multivariable functions, and integration of multi-vector valued functions.
• Teaching QQ group (slides, lecture notes): 949748117

Purpose of the Course:

The aim of this course is to ensure that students are competent in higher mathematics encountered throughout engineering, particularly functions of more than one variable, including both theory and extensive practice.

Requirements by the end of this course:

• apply calculus to parametric functions and vector-valued functions; understand the inner and cross products of vectors, and apply them to lines and planes in space; introduce polar coordinates and related equations;
• calculate partial derivatives, total differential and high-order partial derivatives of multivariable functions; find the derivative of implicit functions;
• explain the concepts of directional derivative and gradient and calculate them in two and three dimensions; apply partial derivatives to find the tangent plane and normal line of a surface; locate extreme values of a multivariable function, both unconstrained and under given conditions, and apply the Lagrange multiplier method;
• describe the meaning of double integrals (Cartesian coordinates and Polar coordinates) and evaluate them; similarly for triple integral (Cartesian coordinates, Cylindrical coordinates and Spherical coordinates);
• explain the definition of line integrals for scalar-valued functions and vector fields, be aware that the results of a line integral depends on the path in general; determine that a line integral is independent of path in conservative fields;
• apply Green’s theorem to integrals in the plane; express given surfaces in an appropriate form and evaluate surface integrals over surfaces; apply the theorems of Green, Gauss and Stokes to line, surface and volume integrals and explain their significance in engineering;
• state what is meant by a sequence and series, find limits of sequence; apply criteria for convergence of series with terms of the same sign or alternating sign; distinguish between absolute convergence and conditional convergence; establish conditions for convergence of power series, other functional series and Taylor series; derive Maclaurin expansions of elementary transcendental functions such as sin(x) and cos(x); apply direct and indirect expansion methods of some simple functions to applications of power series in approximate calculations.

Textbook:

• G. Thomas, M. D. Weir and J. Hass, Thomas’ Calsulus, 13th edition, Pearson, 2016.

Reference:

• 同济大学数学系编,《高等数学》(第五版)，上、下册，高等教育出版社，2001。

Slides

• Practical Information Slide
• Chapter 10. Infinite Sequences and Series Slide
  • 10.1 Sequences
  • 10.2 Infinite Series
  • 10.3 The Integral Test
  • 10.4 Comparison Tests
  • 10.5 Absolute Convergence; The Ratio and Root Tests
  • 10.6 Alternating Series and Conditional Convergence
  • 10.7 Power Series
  • 10.8 Taylor and Maclaurin Series
  • 10.9 Convergence of Taylor Series
  • 10.10 The Binomial Series and Applications of Taylor Series
• Chapter 11. Parametric Equations and Polar Coordinates Slide
  • 11.1 Parametrizations of Plane Curves
  • 11.2 Calculus with Parametric Curves
  • 11.3 Polar Coordinates
  • 11.4 Graphing Polar Coordinate Equations
  • 11.5 Areas and Lengths in Polar Coordinates
  • 11.6 Conic Sections
  • 11.7 Conics in Polar Coordinates
• Chapter 12. Vectors and the Geometry of Space Slide
  • 12.1 Three-Dimensional Coordinate Systems
  • 12.2 Vectors
  • 12.3 The Dot Product
  • 12.4 The Cross Product
  • 12.5 Lines and Planes in Space
  • 12.6 Cylinders and Quadric Surfaces
• Chapter 13. Vector-Valued Functions and Motion in Space Slide
  • 13.1 Curves in Space and Their Tangents
  • 13.2 Integrals of Vector Functions; Projectile Motion
  • 13.3 Arc Length in Space
  • 13.4 Curvature and Normal Vectors of a Curve
  • 13.5 Tangential and Normal Components of Acceleration
  • 13.6 Velocity and Acceleration in Polar Coordinates
• Chapter 14. Partial Derivatives Slide
  • 14.1 Functions of Several Variables
  • 14.2 Limits and Continuity in Higher Dimensions
  • 14.3 Partial Derivatives
  • 14.4 The Chain Rule
  • 14.5 Directional Derivatives and Gradient Vectors
  • 14.6 Tangent Planes and Differentials
  • 14.7 Extreme Values and Saddle Points
  • 14.8 Lagrange Multipliers
  • 14.9 Taylor’s Formula for Two Variables
  • 14.10 Partial Derivatives with Constrained Variables
• Chapter 15. Multiple Integrals Slide
  • 15.1 Double and Iterated Integrals over Rectangles
  • 15.2 Double Integrals over General Regions
  • 15.3 Area by Double Integration
  • 15.4 Double Integrals in Polar Form
  • 15.5 Triple Integrals in Rectangular Coordinates
  • 15.6 Moments and Centers of Mass
  • 15.7 Triple Integrals in Cylindrical and Spherical Coordinates
  • 15.8 Substitutions in Multiple Integrals
• Chapter 16. Integrals and Vector Fields Slide
  • 16.1 Line Integrals
  • 16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux
  • 16.3 Path Independence, Conservative Fields, and Potential Functions
  • 16.4 Green’s Theorem in the Plane
  • 16.5 Surfaces and Area
  • 16.6 Surface Integrals
  • 16.7 Stokes’ Theorem
  • 16.8 The Divergence Theorem and a Unified Theory