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8). , and L denote lengths in the initial and subsequent states, respectively. The deformation of a medium is determined by the displacements of particles and lines of particles. 2. After deformation, the particle is at P ∗ with the position vector R = R(θ1 , θ2 , θ3 ; t) = Xi (θ1 , θ2 , θ3 ; t)ˆıi . 2) asserts that the terminal position of the particle depends on its initial coordinates and time. The continuity of the medium requires that R be continuous. We assume that its derivatives are also continuous.
Again, T ij and Sij , hence Qij are tensorial components. 57) ∂ θ¯i ∂ θ¯j mn ∂θp ∂θq T ij S ij = T Spq . ∂θm ∂θn ∂ θ¯i ∂ θ¯j © 2003 by CRC Press LLC According to the chain rule for partial diﬀerentiation T ij S ij = ∂θp ∂θq mn p q mn T Spq = δm δn T Spq = T mn Smn . 59) In words, this quantity is unchanged by a coordinate transformation. Such quantities are invariants; they have the same value independently of the coordinate system. The invariance hinges on the notions of covariance and contravariance.
51b) serve to deﬁne the covariant derivatives of the contravariant (V i ) and covariant (Vi ) components of a vector. Observe that the covariant derivative (V j |i or Vj |i ) plays the same role as the partial derivative (Vj,i ) plays in the Cartesian coordinate system, that the base vector (g i or g i ) plays the same role as the unit vector ˆıi in the Cartesian system, and that the metric tensor (gij or g ij ) reduces to the Kronecker delta δij in the Cartesian coordinates. 46a), the Christoﬀel symbol Γijk in one coordinate system θ¯i is expressed in terms of the symbols in another system θi by the formula: Γijk = Γlmn ∂θl ∂θm ∂θn ∂θl ∂ 2 θm + glm .