Technology & Engineering
Divergence Theorem
The Divergence Theorem, also known as Gauss's Theorem, is a fundamental theorem in vector calculus. It relates a surface integral over a closed surface to a volume integral over the region enclosed by the surface. In engineering and physics, it is used to relate the flow of a vector field through a closed surface to the divergence of the field within the enclosed volume.
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9 Key excerpts on "Divergence Theorem"
eBook - PDFAll the Math You Missed
(But Need to Know for Graduate School)
- Thomas A. Garrity(Author)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
This method will be illustrated in our sketch of the Divergence Theorem. The second method involves two steps. Step one is to show that given two regions R 1 and R 2 that share a common boundary, we have ∂R 1 function + ∂R 2 function = ∂(R 1 ∪R 2 ) function. Step two is to show that the theorem is true on infinitesimally small regions. To prove the actual theorem by this approach, simply divide the original region into infinitely many infinitesimally small regions, apply step two and then step one. We take this approach in our sketch of Stokes’ Theorem. Again, all of these theorems are really the same. In fact, to most mathematicians, these theorems usually go by the single name “Stokes’ Theorem.” 5.3 A Physical Interpretation of the Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The goal of this section is to give a physical meaning to the Divergence Theorem, which was, in part, historically how the theorem was discovered. We will see that the Divergence Theorem states that the flux of a vector field through a surface is precisely equal to the sum of the divergences of each point of the interior. Of course, we need to give some definitions to these terms. 5.3 A Physical Interpretation of the Divergence Theorem 95 Definition 5.3.1 Let S be a surface in R 3 with unit normal vector field n(x,y,z). Then the flux of a vector field F(x,y,z) through the surface S is S F · n dS . Intuitively we want the flux to measure how much of the vector field F pushes through the surface S . Imagine a stream of water flowing along. The tangent vector of the direction of the water at each point defines a vector field F(x,y,z). Suppose the vector field F is: Place into the stream an infinitely thin sheet of rubber, let us say. We want the flux to measure how hard it is to hold this sheet in place against the flow of the water.
eBook - PDF- William G. Gray, Genetha A. Gray(Authors)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
The time derivative forms are indicative of properties of a region being studied and how this region changes or deforms in time. Here, we seek some general mathematical tools that facilitate the transfer of derivative expressions for the observation of systems between scales. The primary tools that allow such transformations are two theorems: the Divergence Theorem and the transport theorem. Note that these theorems are mathe-matical relations that apply based on the properties of the functions and regions being studied. They are not physical relations. However, when used in conjunction with physical relations, proposed on the basis of observation of system behavior, we are able to obtain transformation of important descriptions of physical problems between scales. Thus the mathematical theorems allow us to “tune up” equations so that they function at the appropriate scale. We will derive these two theorems in turn, in fact based on results of the previous chapters, and then discuss their application to physical systems. 194 195 9.2 Divergence Theorem 9.2 Divergence Theorem The Divergence Theorem relates the values of a quantity at the boundary of the volume to changes in that quantity as a function of position inside the volume. This theorem can be derived most easily making use of Eq. ( 8.56 ), which defines the divergence operator, repeated here for convenience ∇ · f = lim V → 0 1 V f · n d S (9.1) Recall that this equation confirms that the divergence of a function is equal to the net outward flux over the boundary of an infinitesimal volume. To indicate that the volume element, V , and its bounding surface, , are infinitesimally small, we will change to d, eliminating the need to denote explicitly that the volume is approaching infinitesimal smallness, to obtain ∇ · f = 1 d V d f · n d S (9.2) Multiplication of this equation by d V then yields ∇ · f d V = d f · n d S (9.3) We note that d is the boundary surface of the small differential volume.
eBook - PDF- James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION 16.9 The Divergence Theorem 1243 radius a, where a is chosen to be small enough so that S 1 is contained within S. Let E be the region that lies between S 1 and S. Then Equation 7 gives 8 y y E y div E dV - 2 y S1 y E dS 1 y S y E dS You can verify that div E - 0. (See Exercise 25.) Therefore from (8) we have y S y E dS - y S1 y E dS The point of this calculation is that we can compute the surface integral over S 1 because S 1 is a sphere. The normal vector at x is xy | x | . Therefore E n - «Q | x | 3 x S x | x | D - «Q | x | 4 x x - «Q | x | 2 - «Q a 2 since the equation of S 1 is | x | - a. Thus we have y S y E dS - y S1 y E n dS - «Q a 2 y S1 y dS - «Q a 2 AsS 1 d - «Q a 2 4a 2 - 4«Q This shows that the electric flux of E is 4«Q through any closed surface S that contains the origin. [This is a special case of Gauss’s Law (Equation 16.7.11) for a single charge. The relationship between « and « 0 is « - 1 ys4 « 0 d.] ■ Another application of the Divergence Theorem occurs in fluid flow. Let vs x, y, zd be the velocity field of a fluid with constant density . Then F - v is the rate of flow per unit area. If P 0 s x 0 , y 0 , z 0 d is a point in the fluid and B a is a ball with center P 0 and very small radius a, then div FsPd < div FsP 0 d for all points P in B a since div F is con- tinuous. We approximate the flux over the boundary sphere S a as follows: y Sa y F dS - y y Ba y div F dV < y y Ba y div FsP 0 d dV - div FsP 0 dVsB a d This approximation becomes better as a l 0 and suggests that 9 div FsP 0 d - lim a l 0 1 VsB a d y Sa y F dS Equation 9 says that div FsP 0 d is the net rate of outward flux per unit volume at P 0 . (This is the reason for the name divergence.) If div FsPd . 0, the net flow is outward near P and P is called a source.
- James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION 16.9 The Divergence Theorem 1205 radius a, where a is chosen to be small enough so that S 1 is contained within S. Let E be the region that lies between S 1 and S. Then Equation 7 gives 8 y y E y div E dV - 2 y S1 y E dS 1 y S y E dS You can verify that div E - 0. (See Exercise 25.) Therefore from (8) we have y S y E dS - y S1 y E dS The point of this calculation is that we can compute the surface integral over S 1 because S 1 is a sphere. The normal vector at x is xy | x | . Therefore E n - «Q | x | 3 x S x | x | D - «Q | x | 4 x x - «Q | x | 2 - «Q a 2 since the equation of S 1 is | x | - a. Thus we have y S y E dS - y S1 y E n dS - «Q a 2 y S1 y dS - «Q a 2 AsS 1 d - «Q a 2 4a 2 - 4«Q This shows that the electric flux of E is 4«Q through any closed surface S that contains the origin. [This is a special case of Gauss’s Law (Equation 16.7.11) for a single charge. The relationship between « and « 0 is « - 1 ys4 « 0 d.] ■ Another application of the Divergence Theorem occurs in fluid flow. Let vs x, y, zd be the velocity field of a fluid with constant density . Then F - v is the rate of flow per unit area. If P 0 s x 0 , y 0 , z 0 d is a point in the fluid and B a is a ball with center P 0 and very small radius a, then div FsPd < div FsP 0 d for all points P in B a since div F is con- tinuous. We approximate the flux over the boundary sphere S a as follows: y Sa y F dS - y y Ba y div F dV < y y Ba y div FsP 0 d dV - div FsP 0 dVsB a d This approximation becomes better as a l 0 and suggests that 9 div FsP 0 d - lim a l 0 1 VsB a d y Sa y F dS Equation 9 says that div FsP 0 d is the net rate of outward flux per unit volume at P 0 . (This is the reason for the name divergence.) If div FsPd . 0, the net flow is outward near P and P is called a source.
eBook - PDF- James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION 16.9 The Divergence Theorem 1205 radius a, where a is chosen to be small enough so that S 1 is contained within S. Let E be the region that lies between S 1 and S. Then Equation 7 gives 8 y y E y div E dV - 2 y S1 y E dS 1 y S y E dS You can verify that div E - 0. (See Exercise 25.) Therefore from (8) we have y S y E dS - y S1 y E dS The point of this calculation is that we can compute the surface integral over S 1 because S 1 is a sphere. The normal vector at x is xy | x | . Therefore E n - «Q | x | 3 x S x | x | D - «Q | x | 4 x x - «Q | x | 2 - «Q a 2 since the equation of S 1 is | x | - a. Thus we have y S y E dS - y S1 y E n dS - «Q a 2 y S1 y dS - «Q a 2 AsS 1 d - «Q a 2 4a 2 - 4«Q This shows that the electric flux of E is 4«Q through any closed surface S that contains the origin. [This is a special case of Gauss’s Law (Equation 16.7.11) for a single charge. The relationship between « and « 0 is « - 1 ys4 « 0 d.] ■ Another application of the Divergence Theorem occurs in fluid flow. Let vs x, y, zd be the velocity field of a fluid with constant density . Then F - v is the rate of flow per unit area. If P 0 s x 0 , y 0 , z 0 d is a point in the fluid and B a is a ball with center P 0 and very small radius a, then div FsPd < div FsP 0 d for all points P in B a since div F is con- tinuous. We approximate the flux over the boundary sphere S a as follows: y Sa y F dS - y y Ba y div F dV < y y Ba y div FsP 0 d dV - div FsP 0 dVsB a d This approximation becomes better as a l 0 and suggests that 9 div FsP 0 d - lim a l 0 1 VsB a d y Sa y F dS Equation 9 says that div FsP 0 d is the net rate of outward flux per unit volume at P 0 . (This is the reason for the name divergence.) If div FsPd . 0, the net flow is outward near P and P is called a source.
eBook - PDFCalculus
Multivariable
- Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
. . . . . . . . . . . . . . . . . . . 1033 19.3 The Divergence of a Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039 Definition of Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039 Why Do the Two Definitions of Divergence Give the Same Result? . . . . . . . 1041 Divergence-Free Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1042 Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043 19.4 The Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048 The Boundary of a Solid Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048 Calculating the Flux from the Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . 1048 The Divergence Theorem and Divergence-Free Vector Fields . . . . . . . . . . . 1050 Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051 1018 Chapter 19 FLUX INTEGRALS AND DIVERGENCE 19.1 THE IDEA OF A FLUX INTEGRAL Flow Through a Surface Imagine water flowing through a fishing net stretched across a stream. Suppose we want to measure the flow rate of water through the net, that is, the volume of fluid that passes through the surface per unit time. Example 1 A flat square surface of area , in m 2 , is immersed in a fluid. The fluid flows with constant velocity , in m/sec, perpendicular to the square. Write an expression for the rate of flow in m 3 /sec. ✛ Figure 19.1: Fluid flowing perpendicular to a surface Solution In one second a given particle of water moves a distance of ‖ ‖ in the direction perpendicular to the square. Thus, the entire body of water moving through the square in one second is a box of length ‖ ‖ and cross-sectional area .
eBook - PDFCalculus
Early Transcendentals
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
These names originate in the study of fluid flow, in which case the divergence relates to the way in which fluid flows toward or away from a point and the curl relates to the rotational properties of the fluid at a point. We will investigate the physical interpretations of these operations in more detail later, but for now we will focus only on their computation. 15.1.4 DEFINITION If F(x, y, z) = f (x, y, z)i + g(x, y, z) j + h(x, y, z)k, then we de- fine the divergence of F, written div F, to be the function given by div F = ∂ f ∂ x + ∂ g ∂ y + ∂ h ∂ z (7) 15.1 Vector Fields 975 15.1.5 DEFINITION If F(x, y, z) = f (x, y, z)i + g(x, y, z) j + h(x, y, z)k, then we de- fine the curl of F, written curl F, to be the vector field given by curl F = ∂ h ∂ y − ∂ g ∂ z i + ∂ f ∂ z − ∂ h ∂ x j + ∂ g ∂ x − ∂ f ∂ y k (8) REMARK Observe that div F and curl F depend on the point at which they are computed, and hence are more properly written as div F(x, y, z) and curl F(x, y, z). However, even though these functions are expressed in terms of x, y, and z, it can be proved that their values at a fixed point depend only on the point and not on the coordinate system selected. This is important in applications, since it allows physicists and engineers to compute the curl and divergence in any convenient coordinate system. Before proceeding to some examples, we note that div F has scalar values, whereas curl F has vector values (i.e., curl F is itself a vector field). Moreover, for computational purposes it is useful to note that the formula for the curl can be expressed in the determinant form curl F = i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z f g h (9) You should verify that Formula (8) results if the determinant is computed by interpreting a “product” such as (∂ / ∂ x)(g) to mean ∂ g / ∂ x.
- Gary N. Felder, Kenny M. Felder(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
How would this trick apply to other situations? When are you allowed to use it? Why does it work? What is it called? The last question (which you probably weren’t wondering anyway) is easy. We used some- thing called “The Divergence Theorem” to get that formula. For the other answers, you’ll have to read this chapter. 8.2 Scalar and Vector Fields The word “field” has different definitions in different areas of math. For our purposes, a “field” is a function of position. Examples include the temperature in a large room, the magnetic field around the Earth, and the water velocity in the ocean. In many cases the field is changing, so the dependent variable is a function of both position and time. Thissectionintroducessometoolsforvisualizingandworkingwithscalarandvectorfields. 1 For simplicity we will assume there is negligible vertical flow, so you can treat this as a two-dimensional problem. 380 Chapter 8 Vector Calculus 8.2.1 Discovery Exercise: Scalar and Vector Fields The xy-plane is covered with sand. Wandering the plane in your bare feet, you conclude that the depth of sand at any given point can be modeled by the equation z = x 2 sin 2 y. 1. What is the minimum depth of the sand? List three points where you would find this depth. 2. Explain why no point can claim to have the maximum depth of sand. 3. Starting at the point (1, ∕2), you take a long walk in the positive x -direction. Describe what you see (or feel), over time, happening to the sand depth under your feet. See Check Yourself #50 in Appendix L 4. Returning to (1, ∕2) you take another walk, this time in the positive y-direction. Describe your experience during this different walk. Six months later the sand has been swept away by seven maids with seven mops, and has been replaced with a uniform shallow layer of water. Unlike the sand, which was stationary, this water is moving around. The velocity of the water at any point is given by v = x 2 i + 3 j .
- Brian H. Chirgwin, Charles Plumpton(Authors)
- 2016(Publication Date)
- Pergamon(Publisher)
(When Σ is open the discontinuity must tend to zero as P approaches the edge.) (1) Surface divergence We consider an arbitrary closed surface S which encloses a portion, or the whole, of 27. We remove the discontinuity from T by enveloping Σ between two surfaces at a distance apart. (Fig. 23 shows a dia-A I D ^ Σ FIG. 23. §1:7 VECTOR ANALYSIS 75 grammatic section of the arrangement.) The Divergence Theorem, ap-plied to the modified volume T — t, gives / y / d i v a d r = / / a · dS + / / a · άΣ χ + / / a · άΣ 2 , (1.68) τ-t s-t Σ Σ 2 where the last two surface integrals are taken over the surfaces AB, CD respectively, the directions of the vector areas being indicated in Fig. 23. In the limit as ε -> 0 the volume integral, and y y become inte-rs-« grals through T and S respectively, but the remaining integrals tend to finite, non-zero limits. (We assume that the thickness ε is uniform over Σ, although this is not necessary; since diva is finite except on Σ, it can be shown that the limiting values of / / and / f are the same if ε Σ χ Σ, varies over Σ as long as the enveloping surface is regular.) We neglect the contribution from surface integrals over sections such as B G ; such contributions tend to zero as ε -> 0, a being finite. Now / / a · d ^ = / / - (a x ·ί)άΣ + 0(B) , Σχ ΐ'Ι*·άΣ 2 =Ι{(Β 2 ·ί)άΣ+0(ε), Σ* where the integrals on the r. h. sides are taken over that part of Σ inside S. In the limit, as ε -> 0, eqn.
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