ecterrab

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This is about functionality introduced in Maple 2022, which however is still not well known: Integral Vector Calculus and parametrization using symbolic (algebraic) vector notation. Four new commands were added to the Physics:-Vectors package, implementing the parametrization of curves, surfaces and volumes, as well as the computation of path, surface and volume vector integrals. Those are integrals where the integrand is a scalar or vector function. The computation is done from any description (algebraic, parametric, vectorial) of the region of integration - a path, surface or volume.
 
There are three kinds of line or path integrals:

NOTE Jan 1: Updated the worksheet linked below; it runs in Maple 2022.
Download Integral_Vector_Calculus_and_Parametrization.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

This command should have been in Physics on day one. Being more familiar with functional differentiation, and Physics:-Fundiff was the first Physics command that ever existed, I postponed writing LagrangeEquations year after year. In general, however, functional differentiation is seen as a more advanced topic. So there is now a new command, Physics:-LagrangeEquations, taking advantage of functional differentiation on background, and distributed for everybody using Maple 2022.2 within the Maplesoft Physics Updates. This is the first version of its help page.


Download LagrangeEquations.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Vectorial ODEs and vectorial integration constants

In physics, it is traditional to work with vectors, as in analytic geometry, i.e. symbolic vectors, abstract as in `#mover(mi("A"),mo("→"))`, or projected into orthonormal basis such that the unit vectors appear explicitly. In Maple, that is implemented by the Physics:-Vectors  package. The underlying idea is the extension of the Maple computational domain to include a new category of objects: vectors, and related unit vectors and vectorial differential operators all based on `≡`(Nabla, VectorCalculus[Nabla]).

 

But then, with paper and pencil, we frequently write vectorial differential equations, that when solved imply on vectorial integration constants, none of which were implemented; now they are, within the Maplesoft Physics Updates v.1341. As with everything new, there is more work to be done, mainly additional checks for consistency here and there, but the work is advanced; time to tell the story and we are grateful in advance for the always useful opinions / corrections if any.

 

The input/output below illustrate the new features, which by the way compose on top of the new subscripted arbitrary constants by dsolve; this time extended to also be vectorial. The presentation has for context typical material of a first undergrad course in Mechanics. The purpose, anyway, is only to illustrate the new solving of vectorial differential equations and vectorial integration constants.

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Download Vectorial_ODEs_and_integration_constants.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft


 

New display of arbitrary constants and functions

 

When using computer algebra, first we want results. Right. And textbook-like typesetting was not fully developed 20+ years ago. So, in the name of getting those results, people somehow got used to the idea of "give up textbook-quality computer algebra display". But computers keep evolving, and nowadays textbook typesetting is fully developed, so we have better typesetting in place. For example, consider this differential equation:

 

Download New_arbitrary_constants_and_functions.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft


 

New generalized unit vectors in Physics:-Vectors

 

The Physics:-Vectors package, written many years ago to teach Vector Analysis to 1st year undergrad students in Physics courses, introduces several things that are unique in computer algebra software. Briefly, this package has the ability to compute sums, dot and cross products, and differentiation with

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abstract vectors, like `#mover(mi("A"),mo("→"))` or `#mover(mi("A"),mo("→"))`(x, y, z), symbols or functions with an arrow on top that indicates to the system that they are vectors, not scalars;

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projected vectors of algebraic (non-matrix) type in any of the Cartesian, cylindrical or spherical basis and/or associated systems of coordinates, including for that purpose an implementation of the corresponding unit vectors of the three bases;

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abstract or projected differential operators that involve Nabla, Gradient, Divergence, Laplacian and Curl;

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inert vectors or vectorial differential operators, including related expansion of operations and simplification; 

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path, surface and volume vector integrals.

 

In addition to the above, the display is as in textbooks, the input resembles paper and pencil handwriting, and examples of the use of Physics:-Vectors in Vector Analysis are presented in the Physics,Examples page.

 

Download New_generalized_unit_vectors.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

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