Lecture 1 - Lie Groups and the Maurer-Cartan equation
... algebra of left-invariant vector fields on the manifold G. Since this is a Lie subalgebra of the Lie algebra of all differentiable vector fields under the bracket, the Jacobi identity and antisymmetry hold, so we have a lie algebra g canonically associated with the group G, with dim g = dim G. We h ...
... algebra of left-invariant vector fields on the manifold G. Since this is a Lie subalgebra of the Lie algebra of all differentiable vector fields under the bracket, the Jacobi identity and antisymmetry hold, so we have a lie algebra g canonically associated with the group G, with dim g = dim G. We h ...
Studies D
... assessment tasks. All tasks will count toward the school assessment grade. The weightings for assessment items are: approximately 50% for tests, 20% for the examination, while the remaining 30 % is for investigations and/or projects. Time allocations: For a Directed Investigation, a minimum of two ...
... assessment tasks. All tasks will count toward the school assessment grade. The weightings for assessment items are: approximately 50% for tests, 20% for the examination, while the remaining 30 % is for investigations and/or projects. Time allocations: For a Directed Investigation, a minimum of two ...
REVIEW FOR MIDTERM I: MAT 310 (1) Let V denote a vector space
... (a) Complete the following definition: A function T : V −→ W is a linear transformation if ..... (b) Argue directly from the definition in part (a), prove that if T : V −→ P P W is a linear transformation then T ( 3i=1 ai vi ) = 3i=1 ai T (vi ) is true for any real numbers ai and any vectors vi ∈ V ...
... (a) Complete the following definition: A function T : V −→ W is a linear transformation if ..... (b) Argue directly from the definition in part (a), prove that if T : V −→ P P W is a linear transformation then T ( 3i=1 ai vi ) = 3i=1 ai T (vi ) is true for any real numbers ai and any vectors vi ∈ V ...
Document
... The w-coordinate of V determines whether V is a point or a direction vector If w = 0, then V is a direction vector and the fourth column of the transformation matrix has no effect If w 0, then V is a point and the fourth column of the matrix translates the origin ...
... The w-coordinate of V determines whether V is a point or a direction vector If w = 0, then V is a direction vector and the fourth column of the transformation matrix has no effect If w 0, then V is a point and the fourth column of the matrix translates the origin ...
MTH6140 Linear Algebra II 1 Vector spaces
... R2 . This is a real vector space. This means that we can add two vectors, and multiply a vector by a scalar (a real number). There are two ways we can make these definitions. • The geometric definition. Think of a vector as an arrow starting at the origin and ending at a point of the plane. Then add ...
... R2 . This is a real vector space. This means that we can add two vectors, and multiply a vector by a scalar (a real number). There are two ways we can make these definitions. • The geometric definition. Think of a vector as an arrow starting at the origin and ending at a point of the plane. Then add ...
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.