DAVE3700 Mathematics 3000 Course description

In technological and scientific courses, we use mathematics to create models of reality. This enables engineers and natural scientists to calculate the outcome of complicated processes.

Among other things, the course covers mathematics for describing gas and liquid flows in process plants, and air flows in ventilation systems. The methods are also used to describe electromagnetic field propagation in the atmosphere and in conductors. Some of the techniques can be used to calculate the flow volume running through a pipe or watercourse. The Norwegian physicist Vilhelm Bjerknes was a pioneer in the use of this type of mathematics to forecast the weather.

The course deals with subjects that form part of engineering programmes all over the world. An understanding of these subjects will enable students to communicate with other engineers, to participate in professional discussions where the use of mathematics is assumed, and to read specialist literature where mathematics is used. The course also provides a formal background for continued studies leading up to a master’s degree in several fields. The course builds on Mathematics 1000 and Mathematics 2000.

The elective course is initiated provided that a sufficient number of students choose the course.

Passed Mathematics 2000 (all study programmes).

No requirements over and above the admission requirements.

After completing the course, the student is expected to have achieved the following learning outcomes defined in terms of knowledge, skills and general competence:

Knowledge

The student can:

• use the chain rule to calculate d f / d t where f = f (x (t), y (t))
• give a geometric interpretation to the use of the chain rule
• use the substitution method to calculate the largest and / or smallest value of a function under one constraint
• give a geometric description of the idea behind Lagrange's method with one constraint, and be able to use the method
• set up Lagrange's equations when there are multiple constraints

• parametrize a curve in the plane and in space in Cartesian coordinates
• calculate position, speed or acceleration when one of the three is known
• calculate curve length, curvature, tangent vector and normal vector for a curve
• describe a curve in the plane in polar coordinates

• sketch a vector field in the plane
• calculate gradient, divergence and curl
• explain the concept of potential for a gradient field

• determine an expression for the line element d s of a parametrized curve
• calculate the line integral for a scalar and a vector field and interpret the answers
• determine when a vector field is conservative
• use the properties of a conservative field to simplify calculations

• calculate double and triple integrals with given boundaries, and give geometric interpretations of the results
• determine the boundaries of double integrals when the integration region is described in Cartesian coordinates or in polar coordinates
• determine the boundaries of triple integrals when the integration region is described in Cartesian coordinates, cylindrical coordinates or spherical coordinates

• compute using Green's theorem
• use Green's theorem to calculate the circulation of a vector field
• use Green's theorem to derive the divergence theorem in the plane
• calculate the flux of a vector field through a curve
• use the divergence theorem to calculate the flux through closed curves
• explain surface integrals, and be able to calculate surface integrals when it is easy to calculate d S and when the area is the graph of z = f (x, y)
• calculate flux through surfaces when it is easy to calculate and when the surface is the graph of z = f (x, y)
• use the divergence theorem to calculate the flux through closed surfaces
• compute using Stokes' theorem

Skills

The student can:

• discuss the chain rule for a function of two variables, and explain how to determine the largest and / or smallest values ​​for functions of several variables under constraints
• discuss how to describe the movement of particles in the plane and in space
• discuss the concepts of gradient, divergence and curl
• compare line integrals of scalar and vector fields, and discuss the concept of conservative field
• discuss differences and similarities in methods and techniques used to calculate double and triple integrals and be able to interpret the results
• discuss the concept of flux for two- and three-dimensional vector fields, and explain the calculation techniques used to calculate flux.

General competence

The student can:

• based on the theory on functions of one variable, can generalize the knowledge of the derivative as a measure of instantaneous change to functions with several variables
• based on the theory of definite integrals for functions of one variable, can generalize this to the integration of functions with several variables
• evaluate their own and other students' academic work, and formulate written and oral assessments of these works in a scientifically correct and accurate manner
• write precise explanations and reasons for procedures, and demonstrate the correct use of mathematical notation

The teaching is organised as scheduled work sessions. During the work sessions, the students shall practise using the material with which they are presented. Exercises consist of group discussions, individual practice in solving assignments, formulating and solving problems and assessing their own and others’ assignments submitted for weekly assessment.

The students shall learn how to assess their own and others’ scholarly works and formulate assessments of them in such a way that the assessment can serve as advice on further studies. These practical exercises will take place in the scheduled part of the work sessions. Students will therefore carry out weekly assessments of exercises set for the week. Information about how the weekly assessment will take place will be given in the lectures.

The students are required to complete exercises between work sessions. The proposed exercises will be directly linked to the course goals. Self-assessment of answers will give students insight into whether they have achieved the goals.

There are no coursework requirements in this course.

Individual written exam, 3 hours.

The exam result can be appealed.

All printed and written aids.

A handheld calculator that cannot be used for wireless communication or to perform symbolic calculations. If the calculator’s internal memory can store data, the memory must be deleted before the exam. Random checks may be carried out.

Grade scale A-F.

One internal examiner. External examiners are used regularly.