What is recorded here?
For a C^q map from a region of R^m to R^n, Sard proves measure-zero results for the images of points where the derivative has less than maximal rank. If m>n, q≥m−n+1 gives zero n-dimensional Lebesgue measure; if m≤n, C^1 suffices for the rank-deficient image to have zero m-dimensional Hausdorff measure.
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Scientific scope
The original defines critical rank as less than min(m,n); for m<n this is not the modern convention rank<n for every point. Retain the distinct measure dimensions and integer differentiability threshold. The proof uses A. P. Morse's decomposition theorem and Whitney's earlier critical-set example; these are inherited, not newly proved here as Sard's sole inventions. No claim that the critical-point set is null or empty, that arbitrary C^1 maps with m>n suffice, or that the result covers arbitrary Sobolev or infinite-dimensional maps.
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People and systems
Arthur Sard
Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
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English catalogue narrative
Sard: differentiability thresholds for null critical-value sets
For a C^q map from a region of R^m to R^n, Sard proves measure-zero results for the images of points where the derivative has less than maximal rank. If m>n, q≥m−n+1 gives zero n-dimensional Lebesgue measure; if m≤n, C^1 suffices for the rank-deficient image to have zero m-dimensional Hausdorff measure.
Scientific scope
The original defines critical rank as less than min(m,n); for m<n this is not the modern convention rank<n for every point. Retain the distinct measure dimensions and integer differentiability threshold. The proof uses A. P. Morse's decomposition theorem and Whitney's earlier critical-set example; these are inherited, not newly proved here as Sard's sole inventions. No claim that the critical-point set is null or empty, that arbitrary C^1 maps with m>n suffice, or that the result covers arbitrary Sobolev or infinite-dimensional maps.
Reading status
This is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.
People and systems
Arthur Sard
Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.