This work gathers in one place what is pertinent about the connections between metric coefficients, basis- and unit vectors in a four-dimensional relativistic manifold. Some of this material can be found scattered elsewhere; its collection into one place reveals connections that either are not known or are obscure, for example that the metric…

Human-driven carbon emissions have resulted in increased levels of dissolved carbon dioxide in the Earth's oceans. This dissolved carbon dioxide reacts with water to form carbonic acid, which impacts ocean acidity as well as the solubility of carbonate-containing compounds, with far-reaching impacts on marine ecosystems and the human communities…

To a student in introductory physics, using vectors is at best an exercise in bookkeeping. A two-dimensional kinematics problem effectively doubles the number of equations that a student must know, and invites the student to memorize factoids: "The horizontal motion is constant," "Gravity is only in the y-direction," etc. Force…

Basis vectors e[subscript alpha] play a useful role in special and general relativity. In particular they allow an expansion of the vectorial spacetime interval dl along infinitesimal curvilinear coordinate differences dl = e[subscript alpha]d[Xi][superscript alpha]: (thus the definition e[subscript alpha] = dl/d[Xi][superscript alpha]). In this…

We use the Survey of Thermodynamic Processes and First and Second Laws-Long, a research-based survey instrument with 78 items at the level of introductory physics, to investigate introductory and advanced students' difficulties with internal energy, work, and heat transfer. We present analysis of data from 12 different introductory and advanced…

This paper demonstrates how the transcendental number "e" may be arrived at by observing the discharge of a capacitor through a fixed resistor and then modelling the system using a simple step-wise procedure. The experimental phase makes use of the Arduino microcontroller, while simple modelling of the system is carried out by means of…

"Representational transformation diagrams" are used to compare and contrast standard textbook presentations of vector line integrals in undergraduate courses in both mathematics and physics. These presentations are taken as the lower anchor in a learning trajectory. Two principal approaches in the lower division are identified, roughly…

Science, technology, engineering, and mathematics (STEM) education is an economic opportunity for an urban school district in Tennessee. Lack of interest in STEM courses was throughout schools with underrepresented populations. An urban district in Tennessee with few schools implementing STEM initiated instructional practices to increase…

Mathematical reasoning skills are a desired outcome of many introductory physics courses, particularly calculus-based physics courses. Novices can struggle to understand the many roles signed numbers play in physics contexts, and recent evidence shows that unresolved struggle can carry over to subsequent physics courses. Positive and negative…

Owing to the importance of a subject like mathematics in the teaching and learning of science, self-learning often poses a challenge to the educator. The objective of this study is to analyse the enhancement of the textual and the media form of self-learning modules to teach algebra and geometry to eighth graders considering their retention…

In this article, we use a simple lattice model to illustrate the concepts of entropy, free energy, and thermodynamic equilibrium from a microscopic perspective. These physical concepts are based on elementary knowledge of discrete probability distribution and weighted average. Calculations about this model involve basic algebra and counting, which…

We conducted a study with introductory and upper-division level physics students in a Mexican and a Spanish university to learn how students recognize the main characteristics of the electric field in three of its more widely used representations, namely, algebraic notation, vector field plot, and electric field lines, and how the students do…

In 1936, Albert Einstein wrote a brief article where he suggested the possibility that a massive object acted as a lens, amplifying the brightness of a star. As time went by, this phenomenon--known as gravitational lensing--has become a powerful research tool in astrophysics. The simplest and symmetrical expression of a gravitational lens is known…

The damped harmonic oscillator physics problem is a standard oscillation problem in calculus physics for pre-engineering students in a community college. The use of alternative linear algebra eigenvalue stability method is consistent with the upper engineering courses using control theory. The external force and friction damping would become the…

We designed a rubric to assess free-response exam problems in order to compare thinking skills evidenced in exams in classes taught by different pedagogies. The rubric was designed based on Bloom's taxonomy and then used to code exam problems. We have analyzed historical and recent exam problems in both algebra-based and calculus-based exams. In…