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Despite being used in many engineering and scientific areas such as physics and mathematics and often taught in high school, graphical vector addition turns out to be a topic prone to misconceptions in understanding even at university-level…
Computing derivatives of tensor expressions, also known as tensor calculus, is a fundamental task in machine learning. A key concern is the efficiency of evaluating the expressions and their derivatives that hinges on the representation of…
These are general notes on tensor calculus which can be used as a reference for an introductory course on tensor algebra and calculus. A basic knowledge of calculus and linear algebra with some commonly used mathematical terminology is…
Learning to use math in physics involves combining (blending) our everyday experiences and the conceptual ideas of physics with symbolic mathematical representations. Graphs are one of the best ways to learn to build the blend. They are a…
There have been several modifications of how basic calculus has been taught, but very few of these modifications have considered the computational tools available at our disposal. Here, we present a few tools that are easy to develop and…
In a series of ten two-dimensional graphical vector addition questions with varying visual representations, most students stuck to a single solution method, be it correct or incorrect. Changes to the visual representation include placing…
Tensor network methods are taking a central role in modern quantum physics and beyond. They can provide an efficient approximation to certain classes of quantum states, and the associated graphical language makes it easy to describe and…
Visual learners think in pictures rather than words and learn best when they utilize representations based on graphs, tables, charts, maps, colors and diagrams. We propose a new pedagogy for teaching pointers in the C programming language…
This paper provides an introduction to trace diagrams at a level suitable for advanced undergraduates. Trace diagrams are a non-traditional notation for linear algebra. Vectors are represented by edges in a diagram, and matrices by markings…
Geometric algebra is the natural outgrowth of the concept of a vector and the addition of vectors. After reviewing the properties of the addition of vectors, a multiplication of vectors is introduced in such a way that it encodes the famous…
We present a new package, VEST (Vector Einstein Summation Tools), that performs abstract vector calculus computations in Mathematica. Through the use of index notation, VEST is able to reduce three-dimensional scalar and vector expressions…
University students taking introductory physics are generally successful executing mathematical procedures in context, but often struggle with the use of mathematical concepts for sense making. Physics instructors note that their students…
Usually the first course in mathematics is calculus. Its a core course in the curriculum of the Business, Engineering and the Sciences. However many students face difficulties to learn calculus. These difficulties are often caused by the…
Tensors, or multi-linear forms, are important objects in a variety of areas from analytics, to combinatorics, to computational complexity theory. Notions of tensor rank aim to quantify the "complexity" of these forms, and are thus also…
Introduction of curvilinear coordinates might be very convenient in many cases. Theoretically, tensor analysis would be most suited. However, tensor notation can't be used in numerical procedure. For example, the strict discrimination of…
Within the field of numerical multilinear algebra, block tensors are increasingly important. Accordingly, it is appropriate to develop an infrastructure that supports reasoning about block tensor computation. In this paper we establish…
In this paper, we propose a method for importing tensor index notation, including Einstein summation notation, into functional programming. This method involves introducing two types of parameters, i.e, scalar and tensor parameters, and…
Graphics (e.g., figures and charts) are ubiquitous in scientific papers, yet separating graphics from text increases cognitive load in understanding text-graphic connections. Research has found that word-scale graphics, or visual…
Vector-space models, from word embeddings to neural network parsers, have many advantages for NLP. But how to generalise from fixed-length word vectors to a vector space for arbitrary linguistic structures is still unclear. In this paper we…
Student appreciation of a function is enhanced by understanding the graphical representation of that function. From the real graph of a polynomial, students can identify real-valued solutions to polynomial equations that correspond to the…