Abstract
For scientists and engineers, the Laplacian operator is a fundamental tool that has made it possible to carry out important frontier studies involving wave propagation, potential theory, heat conduction, the distribution of stresses in a deformable solid and quantum mechanics. Knowing, understanding, and manipulating the Laplacian operator allows us to tackle complex and exciting physics, chemistry, and engineering problems. In this paper, contained in the Special Issue “Mathematics as the M in STEM Education”, we present an instructional derivation of the Laplacian operator in spherical coordinates. Our derivation is self-contained and employs well-known mathematical concepts used in all science, technology, engineering, and mathematics (STEM) disciplines. Our lengthy but straightforward procedure shows that this fundamental tool in mathematics is not intractable but accessible to anyone who studies any of the STEM disciplines. We consider that this work may be helpful for students and teachers who wish to discuss the derivation of this vital tool from an elementary approach in their courses.
Highlights
It is not difficult to identify that, factually, mathematics is present in each discipline that makes up the STEM acronym
The subject has been extensively treated within the literature, here, we present it with an approach that attempts to reconcile the lack of detail with which most popular books on mathematics and physics address it [22,23], with the “sophistication” of shorter derivations that make use of non-trivial concepts such as the theory of complex variable [24], or the total angular momentum operator [25]
We present a full-fledged derivation of the Laplacian operator in spherical coordinates
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. A good part of the mathematics belonging to science, technology, and engineering (STE) is literally used as a tool to solve specific problems. It is possible to find “more sophisticated” mathematical tools that, in the same way, are present in all science, technology and engineering disciplines, such as in the case of the Laplacian operator. There are many spherical symmetries present in our universe; it is of particular interest to the science, technology, and engineering disciplines to know and understand the Laplacian operator’s form in spherical polar coordinates. The list could go on; what is evident is that knowing, understanding, and manipulating the Laplacian operator provides scientists and engineers with a powerful tool that allows them to tackle interesting problems related to the STEM disciplines. We believe that this work could be helpful for the self-taught student and for academics who wish to present it fully to students whose work involves using this important tool
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
