1 The Lie Algebra of a Lie Group
... Example: A flow is the same as an action of G = R. Example: The Euclidean group E(n) is a Lie group and it acts on Rn (“space”). Example: The Galilei group G(n + 1) is a Lie group and it acts on Rn+1 (“spacetime”). We also saw how G(n + 1) acts on the phase space of a free particle in Rn , X = Rn × ...
... Example: A flow is the same as an action of G = R. Example: The Euclidean group E(n) is a Lie group and it acts on Rn (“space”). Example: The Galilei group G(n + 1) is a Lie group and it acts on Rn+1 (“spacetime”). We also saw how G(n + 1) acts on the phase space of a free particle in Rn , X = Rn × ...
2.2 The n × n Identity Matrix
... (i.e. multiplying any matrix A (of admissible size) on the left or right by In leaves A unchanged). Proof (of Statement 1 of the Theorem): Let A be a n × p matrix. Then certainly the product In A is defined and its size is n × p. We need to show that for 1 ≤ i ≤ n and 1 ≤ j ≤ p,the entry in the ith ...
... (i.e. multiplying any matrix A (of admissible size) on the left or right by In leaves A unchanged). Proof (of Statement 1 of the Theorem): Let A be a n × p matrix. Then certainly the product In A is defined and its size is n × p. We need to show that for 1 ≤ i ≤ n and 1 ≤ j ≤ p,the entry in the ith ...
ppt - Geometric Algebra
... Forms and exterior calculus Working with just the exterior product, exterior differential and duality recovers the language of forms Motivation is that this is the ‘non-metric’ part of the geometric product Interesting development to track is the subject of discrete exterior calculus This has a disc ...
... Forms and exterior calculus Working with just the exterior product, exterior differential and duality recovers the language of forms Motivation is that this is the ‘non-metric’ part of the geometric product Interesting development to track is the subject of discrete exterior calculus This has a disc ...
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.