from vector-calculus with differential forms in (euclidean ones) and differential forms in surfaces till tensor-fields on manifolds or differentiable section into tensor-products of the tangent-bundle of a space
- vectors in r-3 (3d-space) are used in two ways as a geometric tangent objects and/or as a differential operator.
an element of r-3 is a vertical array of scalars which determine a directional derivative:
with this simple device one can answer how much a given definite function in r-3 vary in some particular vector field and shouldn’t necessarely constant, so measure what quantity of variation of to the direction .
Front tim3 to time would be betta try to visualice as an operator
- But we can also want to answer how another vector field varies with respect to , this is:
with this one is compeled to use how the components of vary toward
- : the set of smooth vector field on the open set . This set can be considered as a -module, that is there is an action:
vector calculus and linear algebra were devised by Grassmann-SaintVenant-Gibbs-Heaviside the years 1820-1900 inter,
to give the tensor analysis, used by Einstein to create relativity in 1907 (approx) and Schrödinger quantica in the 1920’s
to detect the next upgrading you gonna need to go just to Heisenberg-Dirac-Cartan circa 1925-1960
the chinesse’s mathemagizian Shiing-Shen Chern around 1975 gaves us the characteristic classes which are the new basic-math tools to unlock the misteries of all maths and sciences…
Evidently, any Bachellor in a CEI (exact science institution) should, at least, be aware that science had grew too high but too systematic too
You could find many Master Science individuals asking for Lie groups and Lie algebras which had be metioned in their graduate ateliers.
To a PhD : ignorating these matters is a MEASURE of their level of culture and of intelligence i.e. competence
today, vector calculus and linear algebra are courses taken by a variety of CEI-students, having teachers of all kinds of culture trying to make them assimilated by their pupils
corruption (tranza-y-media), however time to time one or two (y lo logran de cuando en cuando…) Those who advance will cry mamma (os que avanzan al M.Sc. llorarán a su mamá…) the PhD’ll gonne mad (los PhD enloquecerán…)
Why do not imagine all world to know this techniques? (¿Porqué no inventar querer que todo mundo sepa qué es un algebra de grassmann de un espacio vectorial entonces…?)
S.S. Chern forecasted in brief will see how (Chern pronóstico que en breve veríamos como el concepto de fibrado vectorial y su conexión de Leví-Civita llegaría) should reach (a la escuela de ingeniería… bah), we are at least (apenas lo estamos) wittnessing (atestiwändo):
how the physic’s schools and one-or-two technical centers are getting acquainted!