Primarily serving Boston's Metro-North and Metro-West regions
For all regular and home-schooled students wishing to aim higher — starting with Middle and High School, and on to inquiries deriving from university-level theoretical research in Mathematical Physics.
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Linear Algebra and Vector Spaces, Ordinary and Partial Differential Equations, Integral Equations (regular and singular), Perturbation Techniques and Asymptotics for linear and nonlinear systems, Energy-Based Dynamics and Variational Calculus, Green's Function constructions for canonical and arbitrary spatial domains, Potential Theory and Laplace's Equation and their generalization to handle wave propagation in Fluid Mechanics, Acoustics, Electromagnetics, Optics (geometric or otherwise), and in Vibration-Elasticity Theory for Dispersive and Nondispersive systems, Scattering and Diffraction, Transform Methods, Time- and Frequency-Domain formulations and solutions, Complex Variables, the Wiener-Hopf Technique, Correlation Models and Ensembles, Uncertainty Quantification via generalized Polynomial Chaos (gPC), Statistically Stationary and Non-Stationary Processes and their frequency-wavenumber spectra, Tensors and Differential Geometry.
* Student clients for advanced private teaching are encouraged to submit a sample problem a priori. Our purpose is to maximize each meeting's efficiency so that the student will get the greatest 'bang for his/her buck'. Note also our Mission Statement for tutoring, which follows.
The process of acquiring mathematical intuition involves repeated constructions and deconstructions leading to seemingly tangible visualizations of the previously unimaginable. This exciting metamorphosis also achieves for the individual a kind of cultural and historical connectivity: because it re-enacts humanity's longing for powerful symbols – a longing that each age has only partly satisfied through increasingly higher levels of abstraction.
We'll return to this business of tearing down and rebuilding momentarily. But we first recall a few famous examples of how the history of the development of the symbolic language of mathematical physics has painfully mirrored its learning by nearly everyone who comes to it in its modern form:
(1) Newton gave irrelevant definitions of functional continuity because he lacked the special shorthand of set theory, which finally evoked the needed images;
(2) Euler routinely mangled the product of complex numbers because he inconsistently expressed the imaginary unit in terms of both its present day stand-in letter, 'i' (or 'j'), and its much less transportable original version √-1;
(3) Maxwell's ignorance of the eventual curl operator made his synthesis of Ampere's Law and Faraday's Law into a coherent electrodynamics nearly impossible to apply – though his archaic un-distilled notation had in fact unified electricity, magnetism, and optics;
(4) Lorentz's narrow view of his similarly concealing terminology left him to die believing in an ether medium that his own transformations had rendered superfluous 25 years before;
(5) In the years immediately following Special Relativity (1905), Einstein complained petulantly about how mathematicians had cast his analysis into groups of analogous 4-dimensional vectors. And yet he laboriously went on to master a far more abstract set of equally long (i.e., with 4 components) second-order tensors. The result was his monumental geometrization of gravity in General Relativity (1915);
Now finally about those earlier comments about mathematical constructions and their opposite. We believe that an effective program of one-on-one tutoring in mathematical physics should be based on deliberate reversals in the ultra-condensation of its daunting language. For each short project, or topic, we first consider a set of representative problems through a primitive and down-to-earth vernacular made more natural to the student by her-his initial capacity to visualize. We quickly run through the correspondingly awkward solutions.
We next set the analytical paths to those solutions side by side and 'discover' how their commonality exposes a more economic set of operations that until now had remained hidden – a shorter set of symbols that by their compact form are not only far more convenient, but that also become significant through their capacity to unify, simplify, and thereby illuminate.
Those willing to undergo this imaginative ritual of deconstruction and reconstruction of a mathematical object invariably end up owning its underlying concept. Bizarre abstractions that initially seemed gratuitous now become compelling and effectively touchable. Students feel empowered and look forward to the next challenge. Everything now appears to be within their reach.
Needless to say that the implementation of this basic approach is a strong function of the topic's level, maturity of each student, and other lesser subtleties. Regardless of where you are, we hope that you'll choose it, and us at Top Canonical, in an inevitable and exhilarating reliving of mathematical-physical history – as we tackle together both old and new problems while standing on the shoulders of those linguistically imperfect giants who tried so hard to describe this perfect world.