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Mathematical Preliminaries
Published in R. Ravi, Chemical Engineering Thermodynamics, 2020
The Green’s theorem relates the line integral around a simple closed curve C to a double integral over the region R enclosed by the curve in the x-y plane. The condition for the following relations to hold is that R be a simply connected region. A region R is said to be simply connected [3] if two paths in R with the same end-points can be deformed into each other without moving the end-points and without leaving R. Two forms of the Green’s theorem are: ()∮Cf(x,y)dx=∬R∂f∂ydxdy. ()∮Cg(x,y)dy=−∬R∂g∂xdxdy.
The Axiom of Foundation
Published in C. Young Eutiquio, Vector and Tensor Analysis, 2017
We recall that Green’s theorem expresses a relationship between a line integral on a simple closed curve in the plane and a double integral taken over the plane domain bounded by the curve. The extension of Green’s theorem to three-dimensional space leads to the important divergence theorem or Gauss’ theorem. The theorem relates a surface integral on a closed surface to a volume integral (or triple integral) over the domain bounded by the surface. We state the theorem as follows.
Vector analysis
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
Green’s theorem allowed us to compute a line integral around a closed curve by doing a related double integral over the interior of the curve. In a similar fashion, the divergence theorem (or Gauss’s theorem), allows us to compute surface integrals over surfaces that bound a region by doing a related triple integral over this region. The divergence theorem holds in all dimensions, and its two-dimensional version implies Green’s theorem.
Structural optimization with explicit geometric constraints using a B-spline representation
Published in Mechanics Based Design of Structures and Machines, 2022
Yosef M. Yoely, Iddo Hanniel, Oded Amir
Green’s theorem defines the relation between a line integral around a simple closed curve and the area enclosed by the curve. Therefore we can use Green’s theorem to compute the area A of a closed planar parametric curve