bilinear-differentiability - Di ff erentiating a bilinear...

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Unformatted text preview: Di ff erentiating a bilinear function Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage ∼ squash/ 28 November, 2011 (at 16:20 ) (Below, use VS for vector space , and IPS for inner-product space .) Prolegomenon. In this pamphlet, all VSes are real VSes, R-VSes, as I don’t wish to discuss what C-differ- entiation means. The Product Rule from calculus states: Suppose f , g : R → R are di ff erentiable. Then so is their product, and [ f · g ] = [ f · g ] + [ f · g ] . 1a : It turns out this generalizes. For an R-IPS V : Suppose f , g : R → V are di ff ’able fncs. Then so is h f , g i , and h f , g i = h f , g i + h f , g i . 1b : Also, for Physics problems in 3-dim’al Euclidean space: Suppose f , g : R → R 3 are di ff ’able. Then so is h f , g i . Moreover, h f [ g i = h f [ g i + h f [ g i . 1c : All these raise the question ( not “Beg the question”, which means something di ff erent): What does it...
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