Section 1.15 Solid Mechanics Part III Kelly 1241.15 Tensor Calculus 2: Tensor Functions 1.15.1 Vector-valued functions of a vector Consider a vector-valued function of a vector )(),(jiibaa==baaThis is a function of three independent variables 321,,bbb, and there are nine partial derivatives jiba∂∂/. The partial derivative of the vector awith respect to bis defined to be a second-order tensor with these partial derivatives as its components: jijibaeebba⊗∂∂≡∂∂)((1.15.1) It follows from this that 1−⎟⎠⎞⎜⎝⎛∂∂=∂∂abbaor ijjmmiabbaδ=∂∂∂∂=∂∂∂∂,Iabba(1.15.2) To show this, with )(),(jiijiiabbbaa==, note that the differential can be written as ⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂=∂∂∂∂=∂∂=313212111111abbadaabbadaabbadadaabbadbbadajjjjjjiijjjjSince 321,,dadadaare independent, one may set 032==dada, so that 111=∂∂∂∂abbajjSimilarly, the terms inside the other brackets are zero and, in this way, one finds Eqn. 1.15.2. 1.15.2 Scalar-valued functions of a tensor Consider a scalar valued function of a (second-order) tensor jiijTeeTT⊗==),(φφ. This is a function of nine independent variables, )(ijTφφ=, so there are nine different partial derivatives:
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