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Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952
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Theorem on the differentiation of a composite function with a vector argument; pp. 195–200

(Full article in PDF format) doi: 10.3176/proc.2010.3.01


Authors

Vadim Kaparin, Ülle Kotta

Abstract

The paper provides a theorem on the differentiation of a composite function with a vector argument. The theorem shows how the partial derivative of the total derivative of the composite function can be expressed through the total derivative of the partial derivative of the composite function. The proof of the theorem is based on Mishkov’s formula, which is the generalization of the well-known Faà di Bruno’s formula for a composite function with a vector argument.

Keywords

differential calculus, partial derivative, total derivative, composite function.

References

1. Johnson , W. P. The curious history of Faa di Bruno’s Formula. Amer. Math. Monthly , 2002 , 109 , 217–234.
doi:10.2307/2695352

2. Kaparin , V. and Kotta , Ü. Necessary and sufficient conditions in terms of differential-forms for linearization of the state equations up to input-output injections. UKACC International Conference on CONTROL 2010 7–10 September , Coventry , UK (submitted for publication).

3. Mishkov , R. L. Generalization of the formula of Faa di Bruno for a composite function with a vector argument. Internat. J. Math. Math. Sci. , 2000 , 24(7) , 481–491.
doi:10.1155/S0161171200002970

4. Mishkov , R. L. Nonlinear observer design by reduced generalized observer canonical form. Internat. J. Control , 2005 , 78 , 172–185.
doi:10.1080/00207170500073806
 
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Current Issue: Vol. 68, Issue 4, 2019




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