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Foundation of Electromagnetic Theory
Published in Bahman Zohuri, Patrick J. McDaniel, Electrical Brain Stimulation for the Treatment of Neurological Disorders, 2019
Bahman Zohuri, Patrick J. McDaniel
Now that we have the idea behind the vector divergence operator and its operation, we can then establish the Divergence Theorem. The integral of the divergence of a vector over a volume V is equal to the surface integral of the normal component of the vector over the surface bounding V. That is, () ∫VdivF→dυ=∮SF→⋅n^da
Integrating fields
Published in A.V. Durrant, Vectors in Physics and Engineering, 2019
Surface integrals involve the summation of field values over a surface, an important example being the flux of a vector field across a surface. Flux was introduced in Section 4.5.1 for the special case where the component of the field vector normal to the surface was constant everywhere on the surface. Section 6.5.1 defines flux more generally as a surface integral and introduces other kinds of surface integral. Section 6.5.2 shows how surface integrals are expressed as double integrals and evaluated.
Vector Calculus
Published in John Newman, Vincent Battaglia, The Newman Lectures on Mathematics, 2018
John Newman, Vincent Battaglia
where d-S- $ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{d} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{S} $ represents a surface element vector having the direction of the outward normal to the surface and the surface integral is over the closed surface of the volume element. The net rate of flow of mass out of the volume element is thus obtained by taking the dot product of the mass flux pυ- $ p\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\upsilon } $ with the outward normal unit vector and integrating over the surface of the volume element. On the other hand, the integration of the differential conservation law 13.17 over the volume of the element yields ddt∫pdV=-∫∇·(pυ-)dV. $$ \frac{d}{{dt}}\int {pdV} = - \int {\nabla \cdot (p\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\upsilon } )dV} . $$
Transient energy and exergy analysis of parabolic trough solar collector with an application to Sahel climate
Published in International Journal of Sustainable Energy, 2021
Steven Audrey Ndjanda Heugang, Hervé Thierry Tagne Kamdem, Etienne Tchoffo Houdji, François Beceau Pelap
The equations of the mathematical model of the transient heat-transfer developed above Equations (9), (12) and (15), the boundary and initial conditions Equations (19) and (20) respectively, are solved numerically by using the fully implicit Finite Volume Method (FVM) (Patankar 1980). The HCE is divided into several grids (sub-layers) in axial direction as shown in Figure 2. The grid is uniform such that the sub-layers interface and the domain boundaries coincide with the faces of the control volumes. The principle of this method is to subdivide the computational domain into a set of small ‘juxtaposed’ finite volumes. The centre of each element represents a node. The mesh is such that two separate volumes have one common side. The finite volume method is then used for the integration of the governing equations (Equations (9), (12) and (15), and the boundary and initial conditions) over each control volume. With this method, the first step is the integration of heat-transfer equations, the next, is application of Gauss’s divergence theorem on the conservative term of the equations. The objective of the finite volume method is to obtain a discretised system, keeping in its discretised form the property of energy conservation. Therefore, this theorem transforms volume integrals of divergence terms into surface integrals of fluxes all around the control volume. The heat-transfer equations are then integrated inside these control volumes.
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.
Forced to improve: open book and open internet assessment in vector calculus
Published in International Journal of Mathematical Education in Science and Technology, 2022
The circulation of a vector field is defined as a line integral along a closed curve, being the boundary of the given surface, described in Figure 1. Stokes’ theorem allows the line integral to be substituted with a certain surface integral either over the given surface or any other surface sharing that boundary curve. To answer the question in the example in Figure 1 the student needs to know that the surface can be replaced and then needs to determine the boundaries of the four surfaces described. The question is answered most rapidly if the students already know this information, but they also had access to their textbook, all their notes and any graphing packages available.