By Professor Oscar Gonzalez, Professor Andrew M. Stuart

A concise account of vintage theories of fluids and solids, for graduate and complex undergraduate classes in continuum mechanics.

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Remark: The results in (a) and (b) together imply that the length of a vector and the angle between any two vectors are unchanged by a rotation. The result in (c) implies that rotations commute with the cross product operation; in particular, when two vectors are subject to a common rotation, the normal to their plane is subject to the same rotation. 22 Let Q = I be a rotation tensor. (a) Show that λ = 1 is always an eigenvalue of Q. Hint: Use the characteristic polynomial and properties of determinants.

Since λ = ω we must have e·d = 0. We state, without proof, the following result from linear algebra. 9 Spectral Decomposition Theorem. Let S be a symmetric second-order tensor. Then there exists a right-handed, orthonormal basis {ei } for V consisting of eigenvectors of S. The corresponding eigenvalues λi are the same (up to ordering) for any such basis and form the full set of eigenvalues of S. The dyadic representation of S in any such basis is 3 λi ei ⊗ ei , S= i=1 and the matrix representation is λ1 [S] = 0 0 0 λ2 0 0 0 .

11) We could also deﬁne a change of basis tensor B from {ei } to {ei } by B = Bij ei ⊗ ej where Bij = ei · ej . All that we say for A will also apply to B. However, for convenience, we work only with A. Using the components of A we can express the basis vectors of one frame in terms of the other. For example, a basis vector ej may be expressed in the frame {ei } as ej = (e1 · ej )e1 + (e2 · ej )e2 + (e3 · ej )e3 = (ei · ej )ei , 20 Tensor Algebra e3 e 3/ e 2/ e2 o v e1 e1/ Fig. 4 Two diﬀerent coordinate frames for describing vectors in space.