In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be...

Zerna. Some useful relations in the algebra of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and...

of the coordinates are significant and not the actual values. Some other common coordinate systems are the following: Curvilinear coordinates are a generalization...

relativity requires the use of concepts such as vectors, tensors, pseudotensors and curvilinear coordinates. For an introduction based on the example of particles...

tensor involves the use of superscripted variables (not exponents; see Tensor index notation and Einstein summation notation). The four coordinates of...

vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars...

unchanged by raising and lowering indices. Tensor algebra Tensor calculus Tensors in curvilinear coordinates Rotation group C.W. Misner; K.S. Thorne; J...

the product of tensors if they are seen as multilinear maps (see also tensors as multilinear maps). Thus the components of the tensor product of multilinear...

James Joseph Sylvester in 1851. Curvilinear coordinate systems, such as cylindrical or spherical coordinates, are often used in physical and geometric...

generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra. In differential geometry, an intrinsic[definition...

common curvilinear coordinate systems. This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other...

commutes with a change of (curvilinear) coordinate system. The appropriate expression is more complicated in curvilinear coordinates. The divergence of a vector...

Symmetric tensors occur widely in engineering, physics and mathematics. Let V be a vector space and T ∈ V ⊗ k {\displaystyle T\in V^{\otimes k}} a tensor of...

In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of...

surveying instrument Vector fields in cylindrical and spherical coordinates – Vector field representation in 3D curvilinear coordinate systems Yaw, pitch,...

and covariant derivative are not tensors, the torsion and curvature tensors built from them are. The notation for tensor fields can sometimes be confusingly...

covariant rank n) is sometimes called a permutation tensor. Under the ordinary transformation rules for tensors the Levi-Civita symbol is unchanged under pure...

gradient tensor F {\displaystyle \mathbf {F} } by its transpose. Several rotation-independent deformation gradient tensors (or "deformation tensors", for...

In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with...

specific coordinates requires tensors, capable of connecting such 4‑vectors even within a curved spacetime, and not just within a flat one as in special...

need not be square. Such pure tensors are not generic: if both V and W have dimension greater than 1, there will be tensors that are not pure, and there...

of redirect targets Spherical basis – Basis used to express spherical tensors Halmos, Paul Richard (1987). Finite-Dimensional Vector Spaces (4th ed.)...

operator. In quantum mechanics, physical observables that are scalars, vectors, and tensors, must be represented by scalar, vector, and tensor operators...

Euclidean distance from the origin. In these coordinates, the metric tensor is well-approximated by the Euclidean metric, in the precise sense that g i j =...

is often suppressed, and the notation for tensor fields becomes identical to that for purely algebraic tensors. Over a Riemannian manifold, a metric (field...

symbols are written in the same notation as tensors with index notation, they do not transform like tensors under a change of coordinates. Contracting the...

with respect to each argument. It involves concepts such as matrices, tensors, multivectors, systems of linear equations, higher-dimensional spaces,...

from V using the tensor product and duality. For example, V ⊗ V, the tensor product of V with itself, has a basis consisting of tensors of the form eij...

In mathematics, and particularly topology, a fiber bundle (Commonwealth English: fibre bundle) is a space that is locally a product space, but globally...

{\displaystyle y} -axis. The components of tensors of degree two can be assembled into a matrix. For the inertia tensor this matrix is given by, I = [ I 11 I...