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The gradient, , of a tensor field () in the direction of an arbitrary constant vector c is defined as: = (+) The gradient of a tensor field of order n is a tensor field of order n+1. Cartesian coordinates
For a tensor field of any order k, the gradient = = is a tensor field of order k + 1. For a tensor field of order k > 0, the tensor field of order k + 1 is defined by the recursive relation = where is an arbitrary constant vector.
The divergence of a second-order tensor field in cylindrical polar coordinates can be obtained from the expression for the gradient by collecting terms where the scalar product of the two outer vectors in the dyadic products is nonzero.
The gradient of a function is called a gradient field. A (continuous) gradient field is always a conservative vector field: its line integral along any path depends only on the endpoints of the path, and can be evaluated by the gradient theorem (the fundamental theorem of calculus for line integrals). Conversely, a (continuous) conservative ...
The divergence of a vector field extends naturally to any differentiable manifold of dimension n that has a volume form (or density) μ, e.g. a Riemannian or Lorentzian manifold. Generalising the construction of a two-form for a vector field on R 3, on such a manifold a vector field X defines an (n − 1)-form j = i X μ obtained by contracting ...
A two-dimensional flow that, at the highlighted point, has only a strain rate component, with no mean velocity or rotational component. In continuum mechanics, the strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the strain (i.e., the relative deformation) of a material in the neighborhood of a certain point, at a certain moment of time.
The Laplacian of any tensor field ("tensor" includes scalar and vector) is defined as the divergence of the gradient of the tensor: = (). For the special case where T {\displaystyle \mathbf {T} } is a scalar (a tensor of degree zero), the Laplacian takes on the familiar form.
The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces. Thus, the tangent bundle TM = T(M) might sometimes be written as = = to emphasize that the tangent bundle is the range space of the (1,0) tensor fields (i.e., vector fields) on the manifold M. This should not be confused with the very similar ...