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  2. Einstein notation - Wikipedia

    en.wikipedia.org/wiki/Einstein_notation

    In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity.

  3. Vector calculus identities - Wikipedia

    en.wikipedia.org/wiki/Vector_calculus_identities

    In Einstein notation, ... where the notation ∇ B means the subscripted gradient operates on only the factor B. [1] [2] Less general but similar is the Hestenes ...

  4. List of formulas in Riemannian geometry - Wikipedia

    en.wikipedia.org/wiki/List_of_formulas_in...

    Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise. ... The gradient of ...

  5. Gradient - Wikipedia

    en.wikipedia.org/wiki/Gradient

    The gradient of the function f(x,y) = −(cos 2 x + cos 2 y) 2 depicted as a projected vector field on the bottom plane. The gradient (or gradient vector field) of a scalar function f(x 1, x 2, x 3, …, x n) is denoted ∇f or ∇ → f where ∇ denotes the vector differential operator, del. The notation grad f is also commonly used to ...

  6. Covariance and contravariance of vectors - Wikipedia

    en.wikipedia.org/wiki/Covariance_and_contra...

    In Einstein notation (implicit summation over repeated index), contravariant components are denoted with upper indices as in = A covector or cotangent vector has components that co-vary with a change of basis in the corresponding (initial) vector space. That is, the components must be transformed by the same matrix as the change of basis matrix ...

  7. Tensor derivative (continuum mechanics) - Wikipedia

    en.wikipedia.org/wiki/Tensor_derivative...

    If ,, are the contravariant basis vectors in a curvilinear coordinate system, with coordinates of points denoted by (,,), then the gradient of the tensor field is given by (see [3] for a proof.) = From this definition we have the following relations for the gradients of a scalar field ϕ {\displaystyle \phi } , a vector field v , and a second ...

  8. Two-point tensor - Wikipedia

    en.wikipedia.org/wiki/Two-point_tensor

    Examples include the deformation gradient and the first Piola–Kirchhoff stress tensor. As with many applications of tensors, Einstein summation notation is frequently used. To clarify this notation, capital indices are often used to indicate reference coordinates and lowercase for present coordinates.

  9. Laplace–Beltrami operator - Wikipedia

    en.wikipedia.org/wiki/Laplace–Beltrami_operator

    where here and below the Einstein notation is implied, so that the repeated index i is summed over. The gradient of a scalar function ƒ is the vector field grad f that may be defined through the inner product ⋅ , ⋅ {\displaystyle \langle \cdot ,\cdot \rangle } on the manifold, as