By Murray Gerstenhaber, James D. Stasheff

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There are 12 coordinates in Equations 130 and 131, and therefore, these coordinates, which have physical meaning, can be used to replace the coefficients ai and bi , i = 0, 1, 2, . . , 5 in Equation 129. 132) where e = e (t) = [e1 e2 . . 133) The elements si , i = 1, 2, . . 134) where ξ = x1 /l and η = x2 /l. The procedure described in this section to approximate a field using polynomials and replace the polynomial coefficients using coordinates that can have physical meaning is a fundamental step in the finite element formulation.

147) Substituting these equations into Equation 146, one obtains a set of equations solely expressed in terms of the independent accelerations. 148) In this equation, (BT MB) is the generalized inertia matrix associated with the independent coordinates, and BT (Qe + Qv − MB˙ q˙ i ) is the vector of generalized forces that include applied, centrifugal, and Coriolis forces. The procedure described in this section for writing the dynamic equations in terms of the independent coordinates or the degrees of freedom is called the embedding technique.

A lower-order tensor can be obtained as a special case of Equation 63 by reducing the number of indices. A zero-order tensor is represented by a scalar, a first-order tensor is represented by a vector, and a second-order tensor can be represented by a matrix. A tensor of order n is said to be symmetric with respect to two indices if the interchange of these two indices does not change the value of the elements of the tensor. The tensor is said to be antisymmetric or skew symmetric with respect to two indices if the interchange of these two indices changes only the sign of the elements of the tensor.

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