# MAT-02106 Multivariable Calculus, 5 cr

#### Additional information

The implementation will not be executed during the academic year 2019-2020.

#### Person responsible

Petteri Laakkonen

#### Lessons

Implementation | Period | Person responsible | Requirements |

MAT-02106 2019-01 | - |
Petteri Laakkonen |

#### Learning Outcomes

On this course the students learn to study and visualize the behaviour of a real valued function of two variables using its graphs and contour plots, compute limits of functions of several variables, compute first and higher order partial derivatives, gradients and directed derivatives, and to compute local and global maxima of functions. The students learn to form the derivative matrix of a vector-valued function and to use the chain rule of differentiation. The students know how to compute integrals of real-value functions of two and three variables using cartesian, polar, cylinder and spherical coordinates. The students learn how to justify their claims using mathematical methods and to present their solutions orally as well as in written form.

#### Content

Content |
Core content |
Complementary knowledge |
Specialist knowledge |

1. |
Real valued functions of several variables: Graphs and contour plots of functions of two variables. Limits and continuity, first and higher order partial derivatives, directed derivative and gradient. Linear approximation and differentiability. | The ball, open and closed sets. | Taylor's formula |

2. |
Vector-valued functions of several variables: Derivative matrix and the chain rule. | Hessian matrix | Taylor's formula |

3. |
Extreme values of a function: Local and global maxima and minima. | Constrained extremal values and Lagrange's method. | |

4. |
Integrals of functions of two and three variables: Computing in vertically and horizontally simple domains, computing using polar, cylinder and spherical coordinates. | General change of variables, mean value theorem and average of a function, center of mass and improper integrals. | |

5. |
Using Matlab as a tool in solving the exercise problems. |

#### Prerequisites

Course | Mandatory/Advisable | Description |

MAT-01366 Mathematics 3 | Mandatory | 1 |

MAT-04007 Engineering Mathematics 123 | Mandatory | 1 |

1 . Compulsory prerequisite knowledge.

#### Correspondence of content

Course | Corresponds course | Description |

MAT-02106 Multivariable Calculus, 5 cr | MAT-01466 Mathematics 4, 4 cr |