Abstract
We formalise a statement of Green’s theorem in Isabelle/HOL, which is its first formalisation to our knowledge. The theorem statement that we formalise is enough for most applications, especially in physics and engineering. An interesting aspect of our formalisation is that we neither formalise orientations nor region boundaries explicitly, with respect to the outwards-pointing normal vector. Instead we refer to equivalences between paths.
NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence Program.
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Notes
- 1.
This line integral can be physically interpreted as the work done by F on the \(\partial D\), making this statement a special case of the 3-dimensional Kelvin-Stokes’ theorem. If the line integral is replaced with \(\underset{ \partial D }{\oint } F_x dx - F_y dy\), it can be interpreted as the flux of F through \(\partial D\) and the theorem would be the 2-dimensional special case of the divergence theorem.
- 2.
Using elementary regions that are bounded by \(C^1\) smooth functions is as general as using piece-wise smooth functions because it can be shown that the latter can be divided into regions of the former type (see [9]).
- 3.
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Acknowledgement
This research was supported in part by an Australian National University - International Alliance of Research Universities Travel Grant and by an Australian National University, College of Engineering and Computer Science Dean’s Travel Grant Award. Also, the first author thanks Katlyn Quenzer for helpful discussions.
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Abdulaziz, M., Paulson, L.C. (2016). An Isabelle/HOL Formalisation of Green’s Theorem. In: Blanchette, J., Merz, S. (eds) Interactive Theorem Proving. ITP 2016. Lecture Notes in Computer Science(), vol 9807. Springer, Cham. https://doi.org/10.1007/978-3-319-43144-4_1
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DOI: https://doi.org/10.1007/978-3-319-43144-4_1
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