# A Computational Model for Multi-variable Differential Calculus

• A. Edalat
• A. Lieutier
• D. Pattinson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3441)

## Abstract

We introduce a domain-theoretic computational model for multi-variable differential calculus, which for the first time gives rise to data types for differentiable functions. The model, a continuous Scott domain for differentiable functions of n variables, is built as a sub-domain of the product of n + 1 copies of the function space on the domain of intervals by tupling together consistent information about locally Lipschitz (piecewise differentiable) functions and their differential properties (partial derivatives). The main result of the paper is to show, in two stages, that consistency is decidable on basis elements, which implies that the domain can be given an effective structure. First, a domain-theoretic notion of line integral is used to extend Green’s theorem to interval-valued vector fields and show that integrability of the derivative information is decidable. Then, we use techniques from the theory of minimal surfaces to construct the least and the greatest piecewise linear functions that can be obtained from a tuple of n + 1 rational step functions, assuming the integrability of the n-tuple of the derivative part. This provides an algorithm to check consistency on the rational basis elements of the domain, giving an effective framework for multi-variable differential calculus.

### References

1. 1.
Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D. M., Maibaum, T. S. E., (eds.) Handbook of Logic in Computer Science, vol. 3. Clarendon Press (1994)Google Scholar
2. 2.
Bloomenthal, J. (ed.): Introduction to implicit surfaces. Morgan Kaufmann, San Francisco (1997)
3. 3.
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, Chichester (1983)
4. 4.
Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, Heidelberg (1998)
5. 5.
Edalat, A., Krznarić, M., Lieutier, A.: Domain-theoretic solution of differential equations (scalar fields). In: Proceedings of MFPS XIX. Electronuc Notes in Theoretical Computer Science, vol. 73 (2003), Full paper in http://www.doc.ic.ac.uk/~ae/papers/scalar.ps
6. 6.
Edalat, A., Lieutier, A.: Domain theory and differential calculus (Functions of one variable). Mathematical Structures in Computer Science 14(6), 771–802 (2004)
7. 7.
Edalat, A., Lieutier, A.: Foundation of a computable solid modelling. Theoretical Computer Science 284(2), 319–345 (2002)
8. 8.
Edalat, A., Pattinson, D.: A domain theoretic account of picard’s theorem. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 494–505. Springer, Heidelberg (2004), http://www.doc.ic.ac.uk/
9. 9.
10. 10.
Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)
11. 11.
Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer, Heidelberg (1988)Google Scholar
12. 12.
Rall, L.B., Corliss, G.F.: Automatic differentiation: Point and interval AD. In: Pardalos, P.M., Floudas, C.A. (eds.) Encyclopedia of Optimization, Kluwer, Dordrecht (1999)Google Scholar
13. 13.
Weihrauch, K.: Computable Analysis (An Introduction). Springer, Heidelberg (2000)