MAT531 Geometry/Topology Final Exam Review Sheet Program of
... structure on X, it is enough to apply any homeomorphism X → X that is not a diffeomorphism. A smooth manifold with boundary is defined in almost the same way. We only need to replace Rn with the closed upper half-space. The boundary of X is defined as a set of points x ∈ X that are mapped to the bou ...
... structure on X, it is enough to apply any homeomorphism X → X that is not a diffeomorphism. A smooth manifold with boundary is defined in almost the same way. We only need to replace Rn with the closed upper half-space. The boundary of X is defined as a set of points x ∈ X that are mapped to the bou ...
Definition: A matrix transformation T : R n → Rm is said to be onto if
... origin parallel to the vector (−2, 1) Eigenvalues of powers of a matrix Theorem 5.1.4: If k is a positive integer, λ is an eigenvalue of a matrix A and x is a corresponding eigenvector, then λk is an eigenvalue of Ak and x is a corresponding eigenvector. Idea: Ax = λx ⇒ A2 x = A(Ax) = A(λx) = λAx = ...
... origin parallel to the vector (−2, 1) Eigenvalues of powers of a matrix Theorem 5.1.4: If k is a positive integer, λ is an eigenvalue of a matrix A and x is a corresponding eigenvector, then λk is an eigenvalue of Ak and x is a corresponding eigenvector. Idea: Ax = λx ⇒ A2 x = A(Ax) = A(λx) = λAx = ...
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.