Universal enveloping algebra
... 17.1. Functors. I won’t go through the general definition of categories and functors since we will be working with specific functors not general functors. I will just use vector spaces over a field F , Lie algebras and associative algebras (always with unity) as the main examples. Definition 17.1.1. ...
... 17.1. Functors. I won’t go through the general definition of categories and functors since we will be working with specific functors not general functors. I will just use vector spaces over a field F , Lie algebras and associative algebras (always with unity) as the main examples. Definition 17.1.1. ...
manual - University of Bath
... a second valence coordinate for bending of the same three collinear atoms in the plane perpendicular to this is automatically generated. If the molecule itself is linear (rather than just a subsection of atoms being collinear) then keyword LINE must be specified to ensure the correct number of inter ...
... a second valence coordinate for bending of the same three collinear atoms in the plane perpendicular to this is automatically generated. If the molecule itself is linear (rather than just a subsection of atoms being collinear) then keyword LINE must be specified to ensure the correct number of inter ...
ON SYSTEMS OF DIFFERENTIAL EQUATIONS IN THE SPACE OF
... is its formal solution. But for x 0 it holds that ba(( xx)) . Hence the equation has no local smooth solutions in a neighborhood of x 0 . In this example the point x0 is a singular point of "an infinite type" of the vector field v( x) a( x) dxd . Note that the point x0 is a singular point ...
... is its formal solution. But for x 0 it holds that ba(( xx)) . Hence the equation has no local smooth solutions in a neighborhood of x 0 . In this example the point x0 is a singular point of "an infinite type" of the vector field v( x) a( x) dxd . Note that the point x0 is a singular point ...
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.