Topic outline

  • General

    SPRING TERM 2013-2014

    Detail of this course may be found on the EMU Calculus website:



    Calculus was first invented to meet the mathematical needs of scientists of the sixteenth and seventeenth centuries, needs that mainly mechanical in nature. Nowadays it is a tool used almost everywhere in the modern world to describe change and motion. Its use is widespread in science, engineering, medicine, business, industry, and many other fields. Calculus also provides important tools in understanding functions and has led to the development of new areas of mathematics including real and complex analysis, topology, and non-euclidean geometry.

    The objective of this course is to introduce the fundamental ideas of the differential and integral calculus of functions of several variables.


    Power series, Taylor Polynomials, Taylor Series, Maclaurin series, Binomial series, Lines and planes, Functions of several variables, Limits and Continuity, Partial Differentiation, Chain Rule, Tangent plane, Critical points, Global and Local Extrema, Directional Derivatives, Gradient, Divergence and Curl, Multiple integrals with applications, Triple integrals with applications, Triple integrals in Cylindrical and Spherical coordinates, Line-, Surface- and Volume Integrals, Independence of path, Green’s Theorem, Conservative Vector Fields, Divergence Theorem, Stoke’s Theorem.


    This course provides the mathematical background for engineering students and is very important, for instance, for advanced courses on partial differential equations, probability and statistical analysis or numerical analysis.


    On succesful completion of the course, the students should be able to:

    • find the radius and the interval of convergence of a power series, indicating at which points the series converges absolutely/conditionally;
    • construct Taylor and Maclaurin series for a given function;
    • use Taylor and Maclaurin series for approximation of functions and estimate the error;
    • use power series to calculate limits;
    • understand and apply two and three dimensional Cartesian coordinate system;
    • recognize and classify the equations and shapes of quadratic surfaces;
    • use the properties of vectors and operations with vectors;
    • recognize and construct the equations of lines and planes;
    • operate with vector functions, find their derivatives and integrals, find the arc length;
    • understand and use the concept of a function of several variables, find it’s domain;
    • calculate the limits of multivariable functions and prove the nonexistence of a limit;
    • find partial derivatives using the properties of differentiable multivariable functions and basic rules;
    • apply partial derivatives for finding equations of tangent planes, normal lines, and for extreme values;
    • evaluate double integrals in Cartesian and polar coordinates and triple integrals in Cartesian and cylindrical coordinates;
    • apply multiple integrals for computing areas and volumes;
    • understand and use integration in vector fields;
    • find line integrals and flux using Green’s Theorem;
    • find circulation of a vector field using Stoke’s theorem;
    • use Divergence Theorem to find the flux of a vector field.

    • Topic 2

      Power Series

      Approximation of Functions with Polynomials What is Power Series, Polynomial Approximation, Linear and Quadratic Approximation, Taylor Polynomials, Aproximation with Taylor Polynomials

      Properties of Power Series Convergence of Power Series, Combining Power Series, Differentiating and Integrating Power Series

      Taylor Series Taylor Series of a Function, Convergence of Taylor Series

      Applications of Taylor Series Limits by Taylor Series, Differentiating Power Series, Integrating Power Series 

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      • Topic 3

        Parametric Equations & Polar Coordinates

        Parametric Equations Basic Ideas, Parametric Parabola, Parametric Circle, Parametric Lines, Parametric Equations of Curves

        Polar Coordinates Defining Polar Coordinates, Converting Between Cartesian and Polar Coordinates, Basic Curves in Polar Coordinates

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        • Topic 4

          Vectors and Vector-Valued Functions

          Vectors in the Plane Basic Vector Operations, Scalar Multiplication, Vector Addition and Subtraction, Vector Components, Magnitude, Vector Operations in Terms of Components, Unit Vectors, Properties of Vector Operations

          Vectors in Three Dimension The xyz-Coordinate System, Equations of Simple Planes, Distances in xyz-Space, Equation of a Sphere, Vectors in R3, Magnitude and Unit Vectors

          Dot Products Two Forms of the Dot Product, Properties of Dot Products, Orthogonal Projections

          Cross Products The Cross Product, Properties of the Cross Product

          Lines and Curves in Space Vector-Valued Functions, Lines in Space, Curves in Space

          Calculus of Vector-Valued Functions The Derivative and Tangent Vector, Orientation of Curves: Unit Tangent Vector, Derivative Rules, Higher Derivatives, Integrals of Vector-Valued Functions

          Length of Curves Arc Length, Paths and Trajectories

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          • Topic 5

            Functions of Several Variables

            Planes and Surfaces Equations of Planes, Parallel and Orthogonal Planes, Intersecting Planes

            Graphs and Level Curves Functions of Two Variables, Domain and Range, Graphs of Functions of Two Variables, Level Curves, Functions of More Than Two Variables

            Limits and Continuity Limit of a Funtion of Two Variables, Limits at Boundary Points, Two-Path Test for Nonexistence of Limits, Continuity of Functions of Two Variables

            Partial Derivatives Derivatives With Two Variables, Partial Derivatives, Higher Order Partial Derivatives, Functions of Three Variables

            The Chain Rule The Chain Rule With One Independent Variable, The Chain Rule With Several Independent Variables, Implicit Differentiation

            Directional Derivatives and the Gradient Directional Derivatives, The Gradient Vector, Interpretations of the Gradient, The Gradient and Level Curves, The Gradient in Three Dimensions

            Tangent Planes and Linear Approximation Tangent Planes, Tangent Planes for F(x,y,z) = 0, Tangent Planes for z = f(x,y), Linear Approximation

            Maximum/Minimum Problems Local Maximum/Minimum Values, Critical Points, Second Derivative Test, Saddle Point

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            • Topic 6

              Multiple Integration

              Double Integrals over Rectangular Regions  Volumes of Solids, Iterated Integrals, Average Value

              Double Integrals over General Regions General Regions of Integration, Iterated Integrals, Change of Perspective, Choosing and Changing the Order of Integration, Regions Between Two Surfaces, Finding Area by Double Integrals

              Double Integrals in Polar Coordinates Polar Rectangular Regions, Double Integrals over Polar Rectangular Regions, More General Polar Regions, Double Integrals over More General Polar Regions

              Triple Integrals Triple Integrals in Rectangular Coordinates, Changing the Order of Integration

              Triple Integrals in Cylindrical Coordinates Cylindrical Coordinates, Integration in Cylindrical Coordinates

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              • Topic 7

                 Vector Calculus

                Vector Fields Vector Fields in Two Dimension, Radial Vector Fields in R2, Vector Fields in Three Dimension, Gradient Fields and Potential Functions

                Line Integrals Scalar Line Integrals in the Plane, Scalar Line Integrals in R3, Line Integrals of Vector Fields, Work Integrals, Circulation and Flux of a Vector Field

                Conservative Vector Fields Types Curves and Regions, Test for Conservative Vector Fields, Finding Potential Functions, Fundamental Theorem for Line Integrals and Path Independence, Line Integrals on Closed Curves, Summary of the Properties of Conservative Vector Fields

                Green's Theorem Circulation Form of Green's Theorem, Flux Form of Green's Theorem, Circulation and Flux on More General Regions

                Divergence and Curl The Divergence, The Curl, Summary of the Properties of Conservative Vector Fields

                Surface Integrals Surface Integrals of Scalar-Valued Functions, Surface Integrals on Explicitely Defined Surfaces, Surface Integrals of Vector Fields, Flux Integrals

                Stokes' Theorem Stokes' Theorem, Two Final Notes on Stokes' Theorem

                Divergence Theorem Divergence Theorem

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