Section 1.6 Solid Mechanics Part III Kelly 301.6 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 )(tρρ=. 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 )(tuu=. In this case, the derivative is defined in the usual way, ttttdtdtΔ−Δ+=→Δ)()(lim0uuu, which turns out to be simply the derivative of the coefficients1, iidtdudtdudtdudtdudtdeeeeu≡++=332211Partial derivatives can also be defined in the usual way. For example, if uis a function of the coordinates, ),,(321xxxu, then 1321321101),,(),,(lim1xxxxxxxxxxΔ−Δ+=∂∂→ΔuuuDifferentials of vectors are also defined in the usual way, so that when 321,,uuuundergo increments 332211,,uduuduuduΔ=Δ=Δ=, the differential of uis 332211eeeudududud++=and the differential and actual increment uΔapproach one another as 0,,321→ΔΔΔuuu. 1assuming that the base vectors do not depend on t
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