# MAT-01466 Mathematics 4, 4 cr

Antti Kuusisto

#### Osaamistavoitteet

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.

#### Sisältö

 Sisältö Ydinsisältö Täydentävä tietämys Erityistietämys 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.

#### Oppimateriaali

 Tyyppi Nimi Tekijä ISBN URL Lisätiedot Tenttimateriaali Book Calculus 6e, Early Transcendentials, Matrix Version Edwards & Penney Sections 13.1-13.8, 14.1-14.7 No

#### Esitietovaatimukset

 Opintojakso P/S Selite MAT-01366 Mathematics 3 Mandatory

#### Vastaavuudet

Opintojakso ei vastaan mitään toista opintojaksoa

 Päivittäjä: Kauhanen Janne, 22.10.2018