Vector bundles and torsion free sheaves on degenerations of elliptic
... all vector bundles over an affine line are trivial. Therefore to define a vector bundle over P1 one only has to prescribe its rank r and a gluing matrix M ∈ GL(r, k[z, z −1 ]). Changing bases in free modules over k[z] and k[z −1 ] corresponds to the transformations M 7→ T −1 M S, where S and T are i ...
... all vector bundles over an affine line are trivial. Therefore to define a vector bundle over P1 one only has to prescribe its rank r and a gluing matrix M ∈ GL(r, k[z, z −1 ]). Changing bases in free modules over k[z] and k[z −1 ] corresponds to the transformations M 7→ T −1 M S, where S and T are i ...
MA352_Differential_Geometry_CIIT_VU
... to above definition two translated vectors are same, but in real life two vectors can have different effect if there initial points are different (see picture below). So we need to incorporate the initial point in the definition of vector. ...
... to above definition two translated vectors are same, but in real life two vectors can have different effect if there initial points are different (see picture below). So we need to incorporate the initial point in the definition of vector. ...
Cartesian tensor
In geometry and linear algebra, a Cartesian tensor uses an orthonormal basis to represent a tensor in a Euclidean space in the form of components. Converting a tensor's components from one such basis to another is through an orthogonal transformation.The most familiar coordinate systems are the two-dimensional and three-dimensional Cartesian coordinate systems. Cartesian tensors may be used with any Euclidean space, or more technically, any finite-dimensional vector space over the field of real numbers that has an inner product.Use of Cartesian tensors occurs in physics and engineering, such as with the Cauchy stress tensor and the moment of inertia tensor in rigid body dynamics. Sometimes general curvilinear coordinates are convenient, as in high-deformation continuum mechanics, or even necessary, as in general relativity. While orthonormal bases may be found for some such coordinate systems (e.g. tangent to spherical coordinates), Cartesian tensors may provide considerable simplification for applications in which rotations of rectilinear coordinate axes suffice. The transformation is a passive transformation, since the coordinates are changed and not the physical system.