Abstract
It is made an attempt to discover an explainable plausible reason for the existence of the conditions of optimality for Cartesian vector direction cosines, having importance in energy mechanical engineering, with the help of the multi-optional hybrid functions entropy conditional optimality doctrine. Substantiation is made in terms of the calculus of variations theory with the help of the special hybrid-optional effectiveness functions uncertainty measure, which includes the hybrid functions entropy of the traditional Shannon’s style. In the studied cases, the simplest variational problems solutions, which are the numbers known as the direction cosines of a Cartesian vector, are stipulated by the specified natural logarithmic quadratic forms. It is proposed to evaluate the uncertainty/certainty degree of the magnitude and direction of a Cartesian vector with the use of the objective functional. This is a new insight into the scientific explanation of the well-known dependency derived in another way. The developed theoretical contemplations and mathematical derivations are finalized with a simplest numerical example for the variated value of the multi-optional hybrid function resulting in the objective functional.
Highlights
The subject area studied in energy mechanical engineering is fairly broad and rather diverse [1] – [4]
The presented paper is aimed at revealing the conditions of optimality for Cartesian vector direction cosines with the help of the multi-optional hybrid functions entropy conditional optimality doctrine
It allows stating that the optimized value is the objective functional (1) with taking into consideration the entropy (2) of the squared multi-optional hybrid functions distribution (11). It is discovered the explanations for the conditions of optimality for Cartesian vector direction cosines with the help of the multi-optional hybrid functions entropy conditional optimality doctrine
Summary
The subject area studied in energy mechanical engineering is fairly broad and rather diverse [1] – [4]. The described aspects of the diversity bring to being the multi-alternativeness characterized in the Subjective Analysis Theory [7], with regards to the subjective effectiveness functions, subjective individuals’ preferences functions optimal distributions based upon the preferences entropy (uncertainty) measure conditional extremum principle. The concept of the multi-optional hybrid functions doctrine [8] – [11] uses the objectively existed categories rather than operates with the someone’s subjectively preferred estimations likewise developed in [7]. Aim. The presented paper is aimed at revealing the conditions of optimality for Cartesian vector direction cosines with the help of the multi-optional hybrid functions entropy conditional optimality doctrine
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