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The Multi-Index Notation
Published in M. W. Wong, Partial Differential Equations, 2022
Euclid (fl. 300 BC) was a Greek mathematician best remembered for his Euclidean geometry and the proof of the infinitude of prime numbers. His 13-volume treatise entitled “Elements” contains topics in geometry and number theory. The axiomatic methods in mathematics can be traced back to Euclid's Elements. The Euclidean space ℝn has been and is still among the most important spaces on which modern analysis is built.
Vector Spaces
Published in Jeff Suzuki, Linear Algebra, 2021
For vectors in ℝn, we were able to define the magnitude of a vector by drawing an analogy with Euclidean geometry: If v=〈v1,v2,…,vn〉, then ||v||=v12+v22+…+vn2. “Math ever generalizes,” so perhaps we can define magnitudes in other types of vectors spaces. We define:
Riemannian Metric and Fundamental Tensors
Published in Bhaben Chandra Kalita, Tensor Calculus and Applications, 2019
for aij=δji, it is called the Euclidean metric, and the corresponding space is called the Euclidean space. The geometry developed on the basis of Euclidean metric is called the Euclidean geometry. Here, yi’’s (when expressed in (3.3.1) form) are called the Euclidean coordinates.
Non-traditional assessments to match creative instruction in undergraduate mathematics courses
Published in International Journal of Mathematical Education in Science and Technology, 2023
Mika Munakata, Ceire Monahan, Erin Krupa, Ashwin Vaidya
We co-taught specially designated sections of this course for a National Science Foundation-funded project (Award #1611876) called ‘Engaged Learning through Creativity in Mathematics and Science’. One of the objectives of the project was to reconsider and revise the undergraduate mathematics curriculum to encourage connections between creativity and mathematics. As such, three of the four authors developed instructional modules based on the various characteristics of creativity as described in the literature. The seminal research in this area identifies these characteristics as being inquisitive, unorthodox, motivated and flexible, questioning norms, having aesthetic taste, connecting ideas, seeing similarities (Amabile, 1983; Sternberg & Williams, 2001). The mathematics content focus of our course was non-Euclidean geometry, data, probability, patterns, and symmetry.
Human spatial learning strategies in wormhole virtual environments
Published in Spatial Cognition & Computation, 2023
Christopher Widdowson, Ranxiao Frances Wang
Another important theoretical concept needing clarification is the distinction between Euclidean vs. non-Euclidean geometry and metric vs non-metric space (e.g., Montello, 1992). A metric space satisfies the following properties: 1) the distance from A to B is zero if and only if A and B are the same point; 2) the distance between two distinct points is positive (positivity); 3) the distance from A to B is the same as the distance from B to A (symmetry); and 4) the distance from A to B is less than or equal to the distance from A to B via any third point C (triangle inequality). A Euclidean space is a type of metric space that also satisfies the parallel postulate, therefore a space can be Euclidean, non-Euclidean but metric, or non-metric at all. A spatial representation that does not conform to Euclidean geometry can have violations specific to Euclidean metric (e.g., parallel postulate), or violations of general metric principles that are not specific to Euclidean geometry (e.g., symmetry or triangle inequality). Therefore, it is important to distinguish between Euclidean vs non-Euclidean and metric vs non-metric spaces. When the experimental evidence only involves violation of the general metric properties, it is more appropriate to call it “non-metric” than non-Euclidean, and the theoretical distinction should be referred to as metric vs non-metric instead of Euclidean vs non-Euclidean.
Geometric theory of topological defects: methodological developments and new trends
Published in Liquid Crystals Reviews, 2021
Sébastien Fumeron, Bertrand Berche, Fernando Moraes
Manipulation of heat flux raises intensive research efforts because of the abundant wealth of potential applications, including thermal shielding or stealth of objects, concentrated photovoltaics or thermal information processing (heat-flux modulators, thermal diodes, thermal transistors and thermal memories). These prospects come from the possibility of designing energy paths in a fashion similar to that of light in transformation optics. To do so, the first step is to understand the main peculiarities of heat transfer in the presence of a non-Euclidean geometry. Generally speaking, diffusion of a passive scalar (for instance, the temperature field) can be seen as a collection of Markov processes obeying the stochastic Fokker-Planck equation. In the case of Brownian motion, the Fokker-Planck equation reduces to the well-known parabolic heat equation [206]. When considering diffusion processes in the presence of a non-Euclidean space, the problem is addressed, as already discussed, by replacing the Laplace operator with the Laplace-Betrami operator [207]: Here, D is the diffusivity and its value depends on the material properties. Ought to the form of the metric of a wedge disclination, heat conduction locally occurs as in a monoclinic-like crystal with no internal source [208]: the heat flux vectors are no longer perpendicular to the isothermal surfaces, which are bent depending on the value of the Frank angle. In other words, disclinations in nematics generate thermal lensing effects.