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...Siam j matrix analysis the change of variables formula using volume adi ben israel abstract is a generalization to rectangular matrices absolute value determinant in particular can be used formul instead if jacobi underlying transformation this result applicable integration on surfaces illustrated here by several examples introduction title z f v dv u detj du where are sets rn suciently well behaved function and integrable dx denotes element dxn or jacobian i also denoted n representing derivative an advantage that translated perhaps simpler was given following euler case lagrange it gave prominence functional symbolic determinants e non numerical including functions operators as elements spaces dierent dimensions say rm with m then cannot its present form however full column rank throughout we replace volj get recall r s x vola det ij submatrix rows columns index set nonsingular submatrices see g alternatively product singular values simply detata date february revised may mathematics...