Chemical Engineering Hand Written_Notes_Part_10

Chemical Engineering Hand Written_Notes_Part_10 - cient. In...

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CHAPTER 2 Fundamentals of Functional Analysis 1. Introduction When we begin to use concept of vectors in formulating mathematical models for engineering systems, we tend to associate the concept of a vector space with the three dimensional coordinate space. The three dimensional space we are familiar with can be looked upon as a set of objects called vectors, which satisfy certain generic properties. While working with mathematical modeling, however, we need to deal with variety of such sets containing such objects. It is possible to ’distill’ essential properties satis f ed by vectors in the three dimensional vector space and develop a more general concept of a vector space, which is collection of objects satisfying these properties. Such a generalization can provide a uni f ed view of problem formulations and the solution techniques. Generalization of the concept of vector and vector space to any general set other than collection of vectors in three dimensions is not su
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Unformatted text preview: cient. In order to work with these sets of generalized vectors, we need various algebraic and geometric structures on these sets, such as norm of a vector, angle between two vectors or convergence of a sequence of vectors. To understand why these struc-tures are necessary, consider the fundamental equation that arises in numerical analysis (1.1) F ( x ) = where x is a vector and F ( . ) represents some linear or nonlinear operator, which when operates on x yields the zero vector . In order to generate a numerical approximation to the solution of equation (6.2), this equation is further trans-formed to formulate an iteration sequence as follows (1.2) x ( k +1) = G x ( k ) ; k = 0 , 1 , 2 , ...... where x ( k ) : k = 0 , 1 , 2 , ...... is sequence of vectors in vector space under con-sideration. The iteration equation is formulated in such a way that the solution x of equation ( 1.2 ), i.e. x = G [ x ] 23...
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This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.

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Chemical Engineering Hand Written_Notes_Part_10 - cient. In...

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