# a’lgebra multilineal (with real calculus): synoptic view

remember, here we allow bilingual or trilingual comments,

también mucha geometría :D

 Abstract Multilinear Algebra Duality: covectors Tensor product of  linear transformations Tensor product of  vector spaces Tensor algebra Grassmann algebra covariant and contravariant types Multilinear Algebra  of Inner Product Spaces Reciprocal basis Metric tensor, lenght, area, volumen Bilinear transformations Musical isomorphisms Change of basis, mas Calculus in $\mathbb{R}^n$ Partial derivatives Taylor series Jacobians Chain’s rule Directional derivatives Covariant derivative and Gauss equation Coordinated changes Differential forms with exterior derivatives the $\mathbb{R}^3$ de Rham’s complex Covariant gradient little Stokes’ theorems: Green, Gauss. Algebraic Differential Geometry Parameterizations: curves and surfaces Tangent vectors, tangent space, tangent bundle Curves in $\mathbb{R}^2$ and $\mathbb{R}^3$ and on surfaces in $\mathbb{R}^3$ Surfaces in $\mathbb{R}^3$ all classical surfaces rendered tangent space change of basis vector fields and tensor fields Christoffel’s symbols (connection coefficients) Curvatures (Gaussian, Mean, Principals, Normal and Geodesic) Integration: Gauss-Bonnet, Stokes Manifolds (topological, differential, analytic, anti-analytic, aritmetic,…) Examples: Lie groups and Fiber bundles

Among the many things that I would like to tell is about the role that unfolds modern math in the way we see nature and social  phenomena today, right now

I am always wondering what would it be to have the best techniques and tools to solve problems

Certainly the tools could be too complex

but about the generalization of calculus, geometry and linear algebra, which amalgamate into differential geometry, it isn’t yet that much too complex that a modern learner could be reaching at his-her  early age.

Think, it is about 130 year ago that vector analysis was incorporated to the engineering schools, but 170 years ago that someone thought about the posibility of abstract algebraic-geometric structures

Yes, the grassmann construction

So, do you still going to allow to teach you spoil teachers and habits to reach the tools to do  real advances to the today knowledge?

References: Grassmann

ok?

### 4 responses to “a’lgebra multilineal (with real calculus): synoptic view”

1. jocuri cu bile rosii

Hi, how are you? I hope you do well. I needed to say that I like a’lgebra multilineal (with real calculus): synoptic view | juanmarqz.

2. km qit

http://arxiv.org/PS_cache/math-ph/pdf/0501/0501001v2.pdf

“Undergraduate Lecture Notes in De Rham–Hodge Theory”

Vladimir G. Ivancevic∗ Tijana T. Ivancevic