SF1626 Calculus in Several Variable 7.5 credits

Flervariabelanalys

This is a basic course in DIFFERENTIAL and INTEGRAL CALCULUS for FUNCTIONS of SEVERAL VARIABLES.

  • Educational level

    First cycle
  • Academic level (A-D)

    A
  • Subject area

    Mathematics
    Technology
  • Grade scale

    A, B, C, D, E, FX, F

Course offerings

Autumn 17 CINTE2 for programme students

Autumn 17 CMEDT2 for programme students

Spring 18 CDEPR1 m.fl. for programme students

Spring 18 CELTE1 m.fl. for programme students

Spring 18 CMAST1 m.fl. for programme students

Spring 18 CINEK1 for programme students

  • Periods

    Spring 18 P3 (3.5 credits), P4 (4.0 credits)

  • Application code

    60448

  • Start date

    2018 week: 3

  • End date

    2018 week: 23

  • Language of instruction

    Swedish

  • Campus

    KTH Campus

  • Number of lectures

    21 (preliminary)

  • Number of exercises

    14 (preliminary)

  • Tutoring time

    Daytime

  • Form of study

    Normal

  • Number of places

    No limitation

  • Part of programme

Spring 18 CSAMH1 for programme students

  • Periods

    Spring 18 P3 (1.5 credits), P4 (6.0 credits)

  • Application code

    60450

  • Start date

    2018 week: 3

  • End date

    2018 week: 23

  • Language of instruction

    Swedish

  • Campus

    KTH Campus

  • Number of lectures

    37 (preliminary)

  • Number of exercises

  • Tutoring time

    Daytime

  • Form of study

    Normal

  • Number of places

    No limitation

  • Part of programme

Spring 18 CMAST ITSY1 for programme students

Spring 17 CINEK1 for programme students

Spring 17 CSAMH1 for programme students

Intended learning outcomes

It is important that the student both UNDERSTANDS the mathematical theory and also knows how to APPLY it to concrete problems.

After completing this course with a passing grade the student should be able to

  • Use, explain and apply fundamental concepts and methods of calculus of several variables, especially
    o interpret graphs of functions and level curves/level surfaces and sketch such curves and surfaces in simple cases
    o calculate partial derivatives and use the chain rule for real valued functions and vector valued functions
    o find and classify critical points
    o use Taylor’s formula to approximate functions with polynomials to a desired degree of accuracy
    o use the Jacobian matrix for linear approximation
    o use the gradient to find directional derivatives and be able to explain its relation to level curves/level surfaces
    o solve certain optimization problems, including problems with constraints
    o explain how multiple integrals are defined and how they can be approximated by Riemann sums
    o evaluate certain multiple integrals by iterated integration and by change of variables, in particular by using polar, cylindrical and spherical coordinates
    o explain how integrals can be used to calculate lengths, area, volumes and other items e. g. mass and center of mass.
    o account for the definition of line integrals, surface integrals and flux integrals and evaluate simple instances using parameterization
    o account for and apply Green’s theorem and the divergence theorem
    o explain the concepts of a potential and a conservative vector field and use them in calculations
  • Propose models for applications that can be described by functions of several variables or vector-valued functions, discuss relevance and accuracy of such models and be aware of how mathematical software can be used in calculus of several variables.
  • Read and understand mathematical texts on calculus of several variables and its applications, and communicate mathematical reasoning within this field orally as well as in writing.

For the higher grades the student should also be able to

  • Explain how the Jacobian matrix can be used to decide whether a function is locally invertible.
  • Apply the implicit function theorem.
  • Account for and apply Stokes’ theorem.
  • Calculate limits of functions of several variables and decide whether a limit exists.
  • Account for the concepts of limit, continuity and differentiability of functions of several variables.
  • Solve problems in several steps that require more extensive computations.
  • Generalize and adapt methods to fit into partly new situations.
  • Solve problems that require methods and concepts from several parts of the course.
  • Deduce important formulae and theorems of calculus in several variables.

Course main content

Euclidian n-space. Functions of several variables and vector-valued functions, including the following concepts: Graph, level curve, level surface. Limits and continuity, differentiability, partial derivatives, the chain rule, differentials. Tangent planes and linear approximation. Taylor’s Formula. Gradient and directional derivative. Jacobian matrix and Jacobian determinant. Invertibility and implicitly defined functions. Coordinate changes. Extreme-value problems. Multiple integrals. Line integrals and Green’s theorem. Flux integrals and the divergence theorem. Stokes’ theorem. Applications.

Eligibility

Basic knowledge of calculus in one variable and linear algebra as presented in SF1624 Algebra and Geometry and SF1625 Calculus in One Variable.
Mandatory for first year, can not be read by other students

Literature

Robert A. Adams, Christopher Essex, Calculus - A Complete Course, 8th edition. ISBN 978-0-321-78107-9.

Examination

  • TEN1 - Examination, 7.5, grade scale: A, B, C, D, E, FX, F

Requirements for final grade

Written exam, possibly with the possibility of continuous examination.

Offered by

SCI/Mathematics

Contact

Henrik Shah Gholian (henriksh@kth.se)

Examiner

Hans Thunberg <thunberg@kth.se>

Henrik Shah Gholian <henriksh@kth.se>

Version

Course syllabus valid from: Autumn 2014.
Examination information valid from: Autumn 2007.