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Scalars and Vectors
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
The divergence theorem (also known as Gauss’s theorem or Ostogradsky’s theorem) relates the flux of a vector field through the closed boundary surface of a volume to the volume integral of the divergence of the vector field over the same volume. Given a volume V with a boundary surface S, the theorem states that () ∮Sv→⋅dS→=∮V∇⋅v→dV.
Integral theorems
Published in D.E. Bourne, P.C. Kendall, Vector Analysis and Cartesian Tensors, 2017
In Chapter 4, scalar and vector fields were defined and the properties of gradient, divergence, and curl were discussed; and in Chapter 5 we were concerned with various integrals of scalar and vector fields, and the techniques whereby such integrals are evaluated. The ground has thus been prepared for the two central theorems in vector analysis: (i) the divergence theorem (also called Gauss’s theorem), which relates the integral of a vector field F over a closed surface S to the volume integral of div F over the region bounded by S; and (ii) Stokes’s theorem which relates the integral of a vector field F around a closed curve to the integral of curl F over any open surface S bounded by . In this chapter we shall prove these theorems and some related results.
Deterministic Radiation Transport Methods
Published in Jerry J. Battista, Introduction to Megavoltage X-Ray Dose Computation Algorithms, 2019
Using the divergence theorem (or Gauss’ theorem) from vector calculus, which states that the flux of a vector function through a closed surface is equal to the integral of the divergence of that function over the volume enclosed by that surface, then the surface integral in Equation 6.28 can be converted into a volume integral as follows: () ∮∂Vj→⋅e^ndA=∫V∇→⋅j→dV=∫V∇→⋅υ→ndV=∫Vυ→⋅∇→ndV
A process-based hydrological model for continuous multi-year simulations of large-scale watersheds
Published in International Journal of River Basin Management, 2023
Marcela Politano, Antonio Arenas, Larry Weber
Partial differential conservation Equations (1), (8) and (9) are solved using a finite volume method. Divergence terms in the equations are converted to surface integrals, which can be evaluated as surface fluxes, using the divergence theorem. The resulting system of ordinary differential equations is solved using the library CVODE of SUNDIALS developed at the Lawrence Livermore National Laboratory (Hindmarsh & Serban, 2016). The Backward Differentiation Formulas (BDFs) with Newton iterations recommended for stiff problems are used. A scaled preconditioned GMRES (Generalized Minimal Residual method) solver is used for the solution of the linear system within the Newton corrections.