Demonstration of Gradient and Curl Concept in Functional Analysis

Abstract

This paper first establishes the concept of "one-point variation" and points out the improper formulation in this respect, and takes the view that the functional is infinitely many variables. In this way, the concept of gradient is more naturally introduced in functional analysis than in the usual literature. Thus, the criterion of Bahhoepr's Euler operator (i.e. gradient operator) is intuitively and concisely expounded, which makes it almost self-evident. It is difficult for students majoring in electronics to study the course "Electromagnetic Field and Microwave Technology". This is because the theory of this course is strong, the concept is abstract, and the applied mathematics knowledge is more. Especially, it is necessary to understand and master the concept of gradient and curl involved in field theory and two important integral formulas, namely Gauss formula and Stokes formula. However, there are few steps in the common textbooks and this part of the content, so this paper makes a more detailed introduction in order to reduce the difficulties in learning. After discussing the concept of gradient in depth, this method is extended to include natural boundary conditions. The concept of curl is also discussed, and the general form of operator decomposition theorem is given. Contrasting with elementary calculus, the concept of Two-type line integral in functional analysis is formed, which eliminates the contingency of existing parameters in solving inverse variational problems. Finally, some assumptions about the field theory of functional analysis are given, and the criterion of general derivation operator is derived.