Tensors Explained
As mathematical calculations become more complex, the equations become more burdensome. Scalar notation denotes only magnitude, while a vector represents a magnitude and a direction. In more advanced mathematics, scientists and engineers use tensors for notation. A tensor represents a magnitude and two or more directions, which includes scalar and vector entities. A zero-order tensor is a scalar, a first-order tensor is a vector and a second-order tensor is a matrix.
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Notation
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A general second-order tensor uses a variable with three subscript, which are usually "i," "j" and "k." Some advanced physics or other higher scientific applications may require tensors above second order, but most application only require a second order or below. Each subscript represents the number of dimensions, which is usually three. A zero-order tensor has no subscripts, a first order has one and a second order has two. This pattern continues into the higher-order tensors.
Kronecker Delta
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The Kronecker delta is special tensor that represents the identity matrix. The tensor equals one when "i" equals "j," but it is zero when "i" does not equal "j." This special tensor also has a "sifting" property, whereby a(i) yields a(j). Only the terms where "i" equals "j" are displayed in the graphic since the other terms are all zero by definition.
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Vector Dot Product
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In order to be useful, a tensor has to do more than merely represent some quantity--it must also be capable of representing a computation. One common mathematical operation is the vector dot product, or inner product. This operation comes from a derivation involving the Kronecker delta. The graphic illustrates two Kronecker deltas being multiplied together. The last line of that equation matches the definition of a vector dot product, so a tensor notation usefully depicts a vector dot product.
Alternating Tensor
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The alternating tensor is another special tensor designed to make certain mathematical operations possible. Like the Kronecker delta, the value of the alternating tensor is conditional on its indices, but where the Kronecker delta has two indices, the alternating tensor has three. The zero values allow a cross product operation when combined with two first-order tensors.
Symmetry
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A symmetric tensor is one in which t(i,j) = t(j,i). An anti-symmetric tensor in which t(i, j) = -t(j,i). Any tensor can be expressed as one-half the sum of a symmetric tensor and an anti-symmetric tensor. In three-dimensional (Cartesian) space, any vector can be transformed into an anti-symmetric tensor by using the alternating vector. In the graphic, omega(k) is the vector and R(i,j) is the tensor.
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