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Vectors_Tensors_06_Vector_Calculus_1_Differentiation

# Vectors_Tensors_06_Vector_Calculus_1_Differentiation -...

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Section 1.6 Solid Mechanics Part III Kelly 30 1.6 Vector Calculus 1 - Differentiation Calculus involving vectors is discussed in this section, rather intuitively at first and more formally toward the end of this section. 1.6.1 The Ordinary Calculus Consider a scalar-valued function of a scalar , for example the time-dependent density of a material ) ( t ρ ρ = . The calculus of scalar valued functions of scalars is just the ordinary calculus. Some of the important concepts of the ordinary calculus are reviewed in Appendix B to this Chapter, §1.B.2. 1.6.2 Vector-valued Functions of a scalar Consider a vector-valued function of a scalar , for example the time-dependent displacement of a particle ) ( t u u = . In this case, the derivative is defined in the usual way, t t t t dt d t Δ Δ + = Δ ) ( ) ( lim 0 u u u , which turns out to be simply the derivative of the coefficients 1 , i i dt du dt du dt du dt du dt d e e e e u + + = 3 3 2 2 1 1 Partial derivatives can also be defined in the usual way. For example, if u is a function of the coordinates, ) , , ( 3 2 1 x x x u , then 1 3 2 1 3 2 1 1 0 1 ) , , ( ) , , ( lim 1 x x x x x x x x x x Δ Δ + = Δ u u u Differentials of vectors are also defined in the usual way, so that when 3 2 1 , , u u u undergo increments 3 3 2 2 1 1 , , u du u du u du Δ = Δ = Δ = , the differential of u is 3 3 2 2 1 1 e e e u du du du d + + = and the differential and actual increment u Δ approach one another as 0 , , 3 2 1 Δ Δ Δ u u u . 1 assuming that the base vectors do not depend on t

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