Search results
Results from the WOW.Com Content Network
The Kulkarni–Nomizu product is an important tool for constructing new tensors from existing tensors on a Riemannian manifold. Let A {\displaystyle A} and B {\displaystyle B} be symmetric covariant 2-tensors.
Let r(x) be the position vector of the point x with respect to the origin of the coordinate system. The notation can be simplified by noting that x = r(x). At each point we can construct a small line element dx. The square of the length of the line element is the scalar product dx • dx and is called the metric of the space.
For each point x of γ, the parallel transport of v at x will be a function of x, and can be written as v(x), where v(0) = v. The function v is determined by the requirement that the covariant derivative of v(x) along γ is 0. This is similar to the fact that a constant function is one whose derivative is constantly 0.
A trigonometric number is a number that can be expressed as the sine or cosine of a rational multiple of π radians. [2] Since sin ( x ) = cos ( x − π / 2 ) , {\displaystyle \sin(x)=\cos(x-\pi /2),} the case of a sine can be omitted from this definition.
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. [1] There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found.
The tensors are classified according to their type (n, m), where n is the number of contravariant indices, m is the number of covariant indices, and n + m gives the total order of the tensor. For example, a bilinear form is the same thing as a (0, 2)-tensor; an inner product is an example of a (0, 2)-tensor, but not all (0, 2)-tensors are inner ...
The process involves expressing () = (,) = (,,) in terms of x, y and z and replacing x, y and z with operators V x V y and V z which from vector operator. The resultant operator is hence a spherical tensor operator T ^ m ( l ) {\displaystyle {\hat {T}}_{m}^{(l)}} . ^ This may include constant due to normalization from spherical harmonics which ...
Cartesian tensors are widely used in various branches of continuum mechanics, such as fluid mechanics and elasticity. In classical continuum mechanics , the space of interest is usually 3-dimensional Euclidean space , as is the tangent space at each point.