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We adopt the convention that a repetition of a subscript in a term shall indicate a sum with respect to that subscript.
See Davis R. D., The inverse problem of the calculus of variations in higher space, Transactions of the American Mathematical Society30 (1928), p. 711–713.
See Hadamard, Leçons sur le calcul des variations, Paris 1910,1, p. 319.
See Bolza, Vorlesungen über Variationsrechnung, p. 626.
The nullity of a determinant is its order minus its rank.
Ann-manifoldS is called regular if on it the coordinates of points neighring a given point admit a representation in terms ofn parameters in which not all of the jacobians of the coordinates with respect to then parameters are zero.
Bliss, The transformations of Clebsch in the calculus of variations, Transactions of the American Mathematical Society17 (1916), p. 595.
By the positive and negative type numbers of a real quadratic form, will be understood the number of positive and negative terms respectively in the form when reduced by a real non-singular linear transformation to squared terms only.
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See Bôcher, Leçons sur les Méthodes de Sturm, Paris (1917), Gauthier-Villars et Cie.
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Morse, M. A generalization of the sturm separation and comparison theorems inn-space. Math. Ann. 103, 52–69 (1930). https://doi.org/10.1007/BF01455690
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DOI: https://doi.org/10.1007/BF01455690