# Mathematics for the natural sciences - programme subject in programmes for Specialization in General Studies (MAT3-01)

### Mathematics R2

#### Geometry

The aims of the training are to enable the apprentice to
• perform calculations with three-dimensional vectors that are represented both geometrically and in co-ordinate form
• use and interpret the scalar and vector product in the calculation of distances, angles, area and volume
• use vector calculus to find equation and parameter presentations for lines, plane and spherical surfaces
• calculate longitudinals, angles and areas in bodies limited by plane and spherical surfaces

#### Algebra

The aims of the training are to enable the apprentice to
• find and analyze recursive and explicit formulae for numerical patterns with or without digital means, and implement and present simple proofs linked to these formulae
• implement and give an account of proof by induction
• sum finite series with or without digital means, derive and use the formulae to the sum of the first n members in arithmetic and geometric series, and use this to solve practical problems
• calculate with infinite geometric series with a constant and variable quotients, determine the area of convergence for these series and present the results

#### Functions

The aims of the training are to enable the apprentice to
• simplify and solve linear and quadratic equations in trigonometric expressions by using relations between the trigonometric functions
• derive central functions and use first and second derivatives to elaborate on and discuss such functions
• transform trigonometric expressions of the type a sin kx + b cos kx , and use these to model periodic phenomena
• give an account of the definition of a definite integral as a limit of a sum and an indefinite integral as an anti-derivative
• calculate integrals of the central functions by anti-derivation, substitution, partial fraction decomposition with linear denominators and integration by parts
• interpret the definite integral in models of practical situations and use it to compute plane areas and volumes of rotating bodies
• formulate a mathematical model with the help of central functions on the basis of observed data, process the model and elaborate on and discuss the result and method

#### Differential equations

The aims of the training are to enable the apprentice to
• model practical situations by converting the problem to a differential equation, solving it and interpreting the result
• solve the first order linear and separable differential equations by calculation and give an account of some important areas of application
• solve homogenous second order differential equations and use Newton’s second law to describe free oscillations by periodic functions
• solve differential equations and draw vector diagrams and integral curves, and interpret them using digital tools

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