# A First Course in Continuum Mechanics by Professor Oscar Gonzalez, Professor Andrew M. Stuart

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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**Extra resources for A First Course in Continuum Mechanics**

**Example text**

4 Second-Order Dyadic Products, Bases The dyadic product of two vectors a and b is the second-order tensor a ⊗ b deﬁned by (a ⊗ b)v = (b · v)a, ∀v ∈ V. 3 Second-Order Tensors 15 In terms of components [a ⊗ b]ij , the above equation is equivalent to ∀v ∈ V, [a ⊗ b]ij vj = (bj vj )ai , which implies [a ⊗ b]ij = ai bj . To complete this last step simply take v = e1 , e2 , e3 in turn. Throughout our developments we will move freely from similar expressions, which hold for all v, to the statement with v removed.

6 Special Classes of Tensors To any tensor S ∈ V 2 we associate a transpose S T ∈ V 2 , which is the unique tensor with the property Su · v = u · S T v ∀u, v ∈ V. We say S is symmetric if S T = S and skew-symmetric if S T = −S. A tensor S ∈ V 2 is said to be positive-deﬁnite if it satisﬁes v · Sv > 0 ∀v = 0, and is said to be invertible if there exists an inverse S −1 ∈ V 2 such that SS −1 = S −1 S = I. The operations of inverse and transpose commute, that is, (S −1 )T = (S T )−1 , and we denote the resulting tensor by S −T .

4. By the change of basis tensor from {ei } to {ei } we mean the tensor A deﬁned by A = Aij ei ⊗ ej where Aij = ei · ej . 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.