Mathematics of Physics and Engineering
146 Functions of Several Variables the points with the Q coordinates {c\ ± (a/2),C2 ± (a/2),c 3 ± (a/2)). The volume of this box ...
Curvilinear Coordinate Systems 147 (b) Verify that, in cylindrical coordinates, w-i^ViS+g. ("•«> r dr \ dr ) r^2 d6^2 dz^2 ' ...
148 Functions of Several Variables determinant similar to (3.1.27): curl F^1 hih 2 h 3 hiqx h 2 q 2 h 3 q 3 dqi dq 2 dqz hxFx h ...
Curvilinear Coordinate Systems 149 J(liiQ2,Q3) = hihzha. Try to derive this equality directly by expand- ing the determinant in ...
150 The Three Theorems 3.2 The Three Integral Theorems of Vector Analysis The theorems of Green, Gauss, and Stokes are usually t ...
Gauss's Theorem 151 EXERCISE 3.2.2.B (a) Verify all equalities in (3.2.2). (b) Verify that, in polar coordinates x = r cos 6, y ...
(^152) The Three Theorems and p be the velocity field and the density of a fluid in a region G. In our discussion of divergence, ...
Gauss's Theorem 153 Hint: Note that divF = V^2 (l/r) = 0 ifr ^ 0. If S encloses the origin, then there exists a small sphere cen ...
154 The Three Theorems the flux across dG, we have dM dt HIpudV - ifpv-da; (3.2.7) G 8G recall that the positive direction of th ...
Stokes's Theorem 155 Theorem. Define the vector field F — Pi + Qj + 0k and consider the outside unit normal vector no to dG; sin ...
156 The Three Theorems has a continuous gradient in G. Then the following three conditions are equivalent: (i) The field F has t ...
Laplace's and Poisson's Equations 157 rise meaning of this formula and demonstrating that (3.2.1), (3.2.4), and (3.2.11) all fol ...
158 The Three Theorems The equation V^2 / - g, (3.2.15) where g is a known continuous function, is called Poisson's equation, af ...
Laplace's and Poisson's Equations 159 Theorem 3.2.5 Let G be a domain in R^3. Assume that the boundary dG of G consists of finit ...
160 The Three Theorems (b) If dfi/dn = dfi/dn on S then there exists a real number c so that /i — fi = c everywhere in Gs- EXERC ...
Laplace's and Poisson's Equations 161 cian J.P.G.L. Dirichlet, who did pioneering work in potential theory, the origins of the o ...
162 The Three Theorems Combining (3.2.23) and (3.2.24) yields /// ("V/"^2 + f9) dV = ///(V/ ' VU + 9U) ^ (3-^2 - 25) G G By the ...
Equations in Vacuum 163 3.3 Maxwell's Equations and Electromagnetic Theory Electromagnetic theory, or electromagnetism, is the b ...
(^164) Maxwell's Equations divE = divB = curl E = P_ £o 0; = - 1 dB dt Gauss and Stokes from vector analysis. In the Internation ...
Equations in Vacuum 165 of the charge. Then (3.3.8) becomes S G and equation (3.3.2) follows from Gauss's Theorem (see Theorem 3 ...
«
4
5
6
7
8
9
10
11
12
13
»
Free download pdf