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By A.V.; Kalinichenko, D.F. Bitsadze

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14) √ This notation distinguishes the norm from the absolute value of a scalar, |s| = s2 . A shorthand notation denoting a2 ≡ a · a is sometimes adopted. A vector a satisfying a = 1 is called a unit vector. 15) where θ(a, b) is the angle between vectors a and b and the norm provides a measure for the length of a vector. Two vectors, a and b, that are perpendicular to each other satisfy the condition a · b = 0. An additional important property that can be proven using the three defining properties of an inner product given above is the Schwarz inequality: |a · b| ≤ a b ∀ a, b ∈ Rn d .

Stress involves two spatial directions, the orientation of a plane sectioning a body and a direction in space along which the stress is evaluated. It is therefore a rank 2 or second-order tensor. Tensors of any order are possible. In practice, we will only be dealing with tensors up to fourth order. g. m for mass, v for velocity and σ for stress. In indicial notation, the tensor’s spatial directions are denoted by indices attached to the symbol. Mass has no direction so it has no indices, velocity has one index, stress two, and so on: m, vi , σij .

A 1 = a 3. e. c = 1 a. Multiplying both sides of this equation on the right by a−1 we find c a−1 = (1 a) a−1 . Using the associativity of the operation, the existence of a right inverse element and finally the existence of a right identity element leads to c a−1 = (1 a) a−1 = 1 (a a−1 ) = 1 1 = 1. The last equality (c a−1 = 1) shows that c = a because a−1 is the unique right inverse of a. Substituting this into our starting equation we find 1 a = a, which proves that 1 is the left identity. 3 What is a tensor?

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A Collection of Problems on the Equations of Mathematical Physics by A.V.; Kalinichenko, D.F. Bitsadze


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