Asee peer logo

Coordinate Transforms and Dual Bases: a Teaching Aid for Undergraduate Engineering Students

Download Paper |


2018 ASEE Annual Conference & Exposition


Salt Lake City, Utah

Publication Date

June 23, 2018

Start Date

June 23, 2018

End Date

July 27, 2018

Conference Session

Mathematics Division Technical Session 3

Tagged Division


Tagged Topic


Page Count




Permanent URL

Download Count


Request a correction

Paper Authors


Guenter Bischof Joanneum University of Applied Sciences

visit author page

Throughout his career, Dr. Günter Bischof has combined his interest in science and engineering application. He studied physics at the University of Vienna, Austria, and acquired industry experience as development engineer at Siemens Corporation. Currently he teaches Engineering Mathematics at Joanneum University of Applied Sciences. His research interests focus on automotive engineering, materials physics, and on engineering education.

visit author page


Benjamin Edelbauer Joanneum University of Applied Sciences

visit author page

Benjamin Edelbauer is currently studying Automotive Engineering at the University of Applied Sciences Joanneum in Graz.

visit author page

Download Paper |


When engineering students are introduced to subjects like classical mechanics, elasticity, electricity and magnetism they encounter – occasionally for the first time – tensors in practical applications. Coordinate transformations play evidently an essential part in the understanding of what it means for a quantity to be a tensor. Coordinate systems are chosen as a matter of taste and convenience to make solving a problem as simple as possible. The coordinates are tools that allow the mathematical analysis of engineering problems, but they are not an intrinsic part of these problems. The freedom to choose coordinates for convenience without being dependent on that choice rests in the center of the understanding of tensors.

A complicating issue in students’ understanding is the emergence of co- and contravariance in tensor algebra and calculus. Rectilinear but oblique angled coordinates capture the essence of this duality without the necessity of making use of derivatives. While Cartesian coordinates involve a metric that remains invisible, i.e. the unit matrix, oblique angled coordinates inherently produce non-diagonal matrices representing their metric tensor. Relations between the basis vectors are no longer simple due to the loss of orthonormality. Remedy to this loss of convenience can be found by a second set of basis vectors that is reciprocally orthonormal to the original basis, the dual basis. The representation of a tensorial quantity in the dual bases paves the way for the students’ comprehension of co- and contravariance.

In this paper a computer program is presented that visualizes user-defined and, in general, oblique angled basis vectors, and plots the associated dual basis vectors. The basis vectors can be manipulated interactively by mouse drag and the associated dual basis vectors are modified simultaneously by the program. Position vectors can be inserted and their co- and contravariant components are computed and displayed. In addition, general rotations of a Cartesian coordinate frame, described by Tait-Bryan angles, can be performed.

The visual output of the program can increase and enhance understanding of dual bases and frame rotations and is therefore well suited as a teaching aid. The program is freely available and can be downloaded from our institution’s home page.

Bischof, G., & Edelbauer, B. (2018, June), Coordinate Transforms and Dual Bases: a Teaching Aid for Undergraduate Engineering Students Paper presented at 2018 ASEE Annual Conference & Exposition , Salt Lake City, Utah. 10.18260/1-2--30226

ASEE holds the copyright on this document. It may be read by the public free of charge. Authors may archive their work on personal websites or in institutional repositories with the following citation: © 2018 American Society for Engineering Education. Other scholars may excerpt or quote from these materials with the same citation. When excerpting or quoting from Conference Proceedings, authors should, in addition to noting the ASEE copyright, list all the original authors and their institutions and name the host city of the conference. - Last updated April 1, 2015