# Vector Analysis

• Claus Müller
Chapter
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 155)

## Abstract

As we have already remarked in the introduction, the full discussion of Maxwell’s equations requires a precise formulation of the fundamental operations of vector analysis. The usual definition with the operator
$$\nabla {\mkern 1mu} = {\mkern 1mu} {e_1}{\mkern 1mu} \frac{\partial }{{\partial {\mkern 1mu} {x^1}}}{\mkern 1mu} + {e_2}{\mkern 1mu} \frac{\partial }{{\partial {\mkern 1mu} {x^2}}}{\mkern 1mu} + {\mkern 1mu} {e_3}{\mkern 1mu} \frac{\partial }{{\partial {\mkern 1mu} {x^3}}}$$
leads to difficulties which are not inherent in the nature of the problems but, rather, produced by the choice of the form in which the operator is stated. If we use the ∇-operator in the above form we must postulate that all of the first derivatives exist. Considered alone these derivatives are not characteristic of the vector fields, and their significance arises only because of their appearance in the operations of forming the divergence and the curl. Thus it seems natural to define these processes directly following the physical point of view, and then to formulate the mathematical operations as definitions accordingly.

## Keywords

Vector Field Surface Element Regular Point Vector Analysis Usual Definition
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