Book contents
- Frontmatter
- Contents
- Preface
- I MATRIX THEORY
- III VECTORS AND TENSORS
- 4 Vector and Tensor Algebra and Calculus
- 5 Vector Integral Theorems
- III ORDINARY DIFFERENTIAL EQUATIONS
- IV PARTIAL DIFFERENTIAL EQUATIONS
- A Additional Details and Fortification for Chapter 1
- B Additional Details and Fortification for Chapter 2
- C Additional Details and Fortification for Chapter 3
- D Additional Details and Fortification for Chapter 4
- E Additional Details and Fortification for Chapter 5
- F Additional Details and Fortification for Chapter 6
- G Additional Details and Fortification for Chapter 7
- H Additional Details and Fortification for Chapter 8
- I Additional Details and Fortification for Chapter 9
- J Additional Details and Fortification for Chapter 10
- K Additional Details and Fortification for Chapter 11
- L Additional Details and Fortification for Chapter 12
- M Additional Details and Fortification for Chapter 13
- N Additional Details and Fortification for Chapter 14
- Bibliography
- Index
5 - Vector Integral Theorems
from III - VECTORS AND TENSORS
Published online by Cambridge University Press: 05 April 2013
- Frontmatter
- Contents
- Preface
- I MATRIX THEORY
- III VECTORS AND TENSORS
- 4 Vector and Tensor Algebra and Calculus
- 5 Vector Integral Theorems
- III ORDINARY DIFFERENTIAL EQUATIONS
- IV PARTIAL DIFFERENTIAL EQUATIONS
- A Additional Details and Fortification for Chapter 1
- B Additional Details and Fortification for Chapter 2
- C Additional Details and Fortification for Chapter 3
- D Additional Details and Fortification for Chapter 4
- E Additional Details and Fortification for Chapter 5
- F Additional Details and Fortification for Chapter 6
- G Additional Details and Fortification for Chapter 7
- H Additional Details and Fortification for Chapter 8
- I Additional Details and Fortification for Chapter 9
- J Additional Details and Fortification for Chapter 10
- K Additional Details and Fortification for Chapter 11
- L Additional Details and Fortification for Chapter 12
- M Additional Details and Fortification for Chapter 13
- N Additional Details and Fortification for Chapter 14
- Bibliography
- Index
Summary
In this chapter, we discuss the major integral theorems that are used to develop physical laws based on integrals of vector differential operations. The general theorems include the divergence theorem, the Stokes' theorem, and various lemmas such as the Green's lemma.
The divergence theorem is a very powerful tool in the development of several physical laws, especially those that involve conservation of physical properties. It connects volume integrals with surface integrals of fluxes of the property under consideration. In addition, the divergence theorem is also key to yielding several other integral theorems, including the Green's identities, some of which are used extensively in the development of finite element methods.
Stokes' theorem involves surface integrals and contour integrals. In particular, it relates curls of velocity fields with circulation integrals. In addition to its usefulness in developing physical laws, Stokes' theorem also offers a key criteria for path independence of line integrals inside a given region that can be determined to be simply connected. We discuss how to determine whether the regions are simply connected in Section 5.3.
In Section 5.5, we discuss the Leibnitz theorems involving the derivative of volume integrals in both 1D and 3D space with respect to a parameter α in which the boundaries and integrands are dependent on the same parameter α These are important when dealing with time-dependent volume integrals.
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- Methods of Applied Mathematics for Engineers and Scientists , pp. 204 - 232Publisher: Cambridge University PressPrint publication year: 2013