# multilinear algebra 1, a synoptic view

what is math? let us discuss:

 Baby Abstract Multilinear Algebra Baby Multilinear Algebra  of Inner Product Spaces Reciprocal basis Metric tensor, lenght, area, volumen Bilinear transformations Musical isomorphisms Change of basis 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) Vector Fields, Covector Fields, Tensor Fields Integration: Gauss-Bonnet, Stokes Baby Manifolds (topological, differential, analytic, anti-analytic, aritmetic,…) Examples: Lie groups and Fiber bundles