{'fraction_non_alphanumeric': 0.028505392912172575, 'fraction_numerical': 0.0030816640986132513, 'mean_word_length': 4.216867469879518, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Differential of Variational Forms

where S is a Borel subset of R', L is a first order differential operator and Lf > 0, f belongs to a continuously differentiable class of functions Y, and ,u is a positive bounded regular Borel measure in Rn. In a large class of problems in continuum mechanics, we wish to determine I(,u) = minfJ I(f, It). The measure It is usually associated with some physical characteristics of the continuum. For example, the natural frequency of an earth model may be obtained by minimizing a form similar to (1) in which pu is the rigidity of the earth model and depends upon the elastic constants of the earth. In this problem we may want to determine the effects of changes in the elastic constants upon the natural frequency of the model. In many other problems, the measure ,u must be so determined as to yield a specified performance which is related to Iu by an integral of the form (1). This is the case in many optimum design problems in structural mechanics. In the problems considered a quantity of interest is the variation of I with respect to changes in Iu. We shall show that if certain conditions are imposed and if fo is the point in F at which I(fo, go) = I(jo), then the derivative of I with respect to gu0 is a function defined on S, and at p E S becomes