Metric tensor

In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space.

Related Topics:
Mathematics - Riemannian geometry - Tensor - Rank - Distance - Angle

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Once a local coordinate system x^i is chosen, the metric tensor appears as a matrix, conventionally denoted G. The notation g_{ij} is conventionally used for the components of the metric tensor (i.e. the elements of the matrix). In the following, we use the Einstein notation for implicit sums.

Related Topics:
Matrix - Einstein notation

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The length of a segment of a curve parameterized by t, from a to b, is defined as:

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:L = int_a^b sqrt{ g_{ij}{dx^iover dt}{dx^jover dt}}dt

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The angle heta between two tangent vectors, U=u^i{partialover partial x_i} and V=v^i{partialover partial x_i} , is defined as:

Related Topics:
Tangent - Vector

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:

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cos heta = rac{g_{ij}u^iv^j}

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The induced metric tensor for a smooth embedding of a manifold into Euclidean space can be computed by the formula

Related Topics:
Embedding - Manifold - Euclidean space

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:G = J^T J

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where J denotes the Jacobian of the embedding and J^T its transpose.

Related Topics:
Jacobian - Transpose

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