Math 5BI: Problem et 9 Integral Theorems of Vector Calculus June 2, 2010 A. ivergence and Curl The gradient operator = i + y j + z k operates not only on scalar-valued functions f, yielding the gradient f = f i + f y j + f z k, but it also acts on vector fields in two different ways. If F(x, y, z) = P (x, y, z)i + Q(x, y, z)j + R(x, y, z)k is a smooth vector field, its divergence is the function ( F = i + y j + ) z k (P i + Qj + Rk) = P + Q y + R z, while its curl is the vector field F = = / y Q = / z R i + / z R ) i + ( R y Q z i j k / / y / z P Q R ( P z R / P j + / P ) j + ( Q P y / y Q k ) k. The geometrical and physical interpretations of the divergence and the curl come from the divergence theorem and tokes s theorem. To understand the divergence theorem, we need to first study the notion of flux integral. uppose that we have a continuous choice of unit-normal N to a smooth 1
surface. uch a continuous choice of unit-normal is called an orientation of. If F(x, y, z) = P (x, y, z)i + Q(x, y, z)j + R(x, y, z)k is a smooth vector field on R 3 the flux of F through is given by the surface integral F NdA. At first, one might think that calculation of flux integrals is difficult. But calculation of flux integrals is simpler than might be expected, because u NdA = v u v dudv = u v dudv, and hence u v F NdA = ( F(x(u, v), y(u, v), z(u, v)) u ) dudv. v Given a parametrization X :, the integral on the right is often rather routine. Problem 9.1. If is the hemisphere x 2 + y 2 + z 2 = a 2, z 0, and evaluate F(x, y, z) = yi xj + z 3 k, F NdA, where N is the upward-pointing unit normal. A physical picture for the flux integral: uppose that a fluid is flowing throughout (x, y, z)-space with velocity V(x, y, z) and density ρ(x, y, z). In this case, fluid flow is represented by the vector field and the surface integral F = ρv, F NdA (1) represents the rate at which the fluid is flowing accross in the direction of N. Indeed, the rate at which fluid flows across a small piece of of surface area da is (density)(normal component of velocity)da = ρv NdA. If we add up the contributions of all the small area elements, we obtain the integral (1). The ivergence Theorem. Let be a region in (x, y, z)-space which is bounded by a piecewise smooth surface. Let N be the outward-pointing unit 2
normal to. If F(x, y, z) is a vector field which is smooth on and its boundary, then ( F)dxdydz = F NdA. The proof of the divergence theorem is very similar to the proof of Green s theorem. Like Green s theorem, the divergence theorem can be used to reduce a complicated surface integral to a simpler volume integral, or a complicated volume integral to a simpler surface integral. Problem 9.2. a. Find the divergence of the vector field F(x, y, z) = log(y 2 + z 2 + 1)i + yj + (sin x cos y)k. b. Use the divergence theorem to evaluate the flux integral F NdA, where is the boundary of the cube 1 x 1, 1 y 1, 1 z 1, and N is the outward-pointing unit normal. Problem 9.3. a. Find the divergence of the vector field F(x, y, z) = xi + yj + zk. b. Let be the unit sphere defined by the equation x 2 + y 2 + z 2 = 1. how that the evaluation of F at a point of the unit sphere is the outward=pointing unit normal to the sphere. how that F NdA = area of. c. Use the divergence theorem to evaluate ( F)dxdydz, where is the unit ball bounded by the unit sphere x 2 + y 2 + z 2 = 1. Use your result to calculate the volume of the unit ball. igression: The equations of fluid mechanics. The reason the divergence theorem is so important is that it can be used to derive many of the important partial differential equations of mathematical physics. For example, we can use it to derive the equation of continuity from fluid mechanics. 3
Indeed, suppose that a fluid is flowing throughout (x, y, z)-space with velocity V(x, y, z, t) and density ρ(x, y, z, t). If we represent the fluid flow by the vector field F = ρv, then the surface integral F NdA represents the rate at which the fluid is flowing accross in the direction of N. We assume that no fluid is being created or destroyed. Then the rate of change of the mass of fluid within is given by two expressions, (x, y, z, t)dxdydz t and F NdA. It follows from the divergence theorem that the second of these expressions equals F(x, y, z, t)dxdydz. Thus t dxdydz = Fdxdydz. ince this equation must hold for every region in (x, y, z)-space, we conclude that the integrands must be equal, t = F = (ρv). Thus we obtain the equation of continuity, + (ρv) = 0. (2) t Actually, the equation of continuity is only one of the important Euler equations or the Navier-tokes equations which are used to model a perfect fluid in fluid mechanics. The Navier-tokes equations are partial differential equations much studied by mathematicians, engineers and physicists. One of the seven millennium prize problems proposed by the Clay Mathematics Institute asks for a proof that solutions to the Navier-tokes equations for given initial conditions exist for all time. ee http://www.claymath.org/millennium/ The other major integral theorem from vector calculus is: 4
tokes s Theorem. If is an oriented smooth surface in R 3 bounded by a piecewise smooth curve, and F is a smooth vector field on R 3, then ( F) NdA = F Tds, where N is the unit normal chosen by the orientation and T is the unit tangent to chosen so that N T points into. tokes s theorem can be used in two directions, to reduce a complicated line integral to a simpler surface integral or a complicated surface integral to a simpler line integral. Problem 9.4. a. Find a parametrization for the circle C of radius one lying in the plane z = 4 and centered on the line x = y = 0, oriented so that it is traversed counterclockwise when viewed from above. b. Find the curl of the vector field F(x, y, z) = yi + xj + 12k. c. Use tokes s theorem to evaluate the line integral F dx. C Remark. tokes s theorem gives rise to a geometric interpretation of curl. Indeed, let (x 0, y 0, z 0 ) be a given point in R 3, N a unit-length vector located at (x 0, y 0, z 0 ), Π the plane through (x 0, y 0, z 0 ) which is perpendicular to N. Let ɛ be the disk in Π of radius ɛ centered at (x 0, y 0, z 0 ), ɛ the circle in Π of radius ɛ centered at (x 0, y 0, z 0 ). Then ( F)(x 0, y 0, z 0 ) N = lim ɛ ( F) NdA ɛ 0 Area of ɛ = lim ɛ F Tds Rate of circulation of F about ɛ = lim. ɛ 0 Area of ɛ ɛ 0 Area of ɛ Thus ( F)(x 0, y 0, z 0 ) measures the rate of circulation of the fluid flow represented by F near (x 0, y 0, z 0 ). B. Maxwell s Equations In 1864, James Clerk Maxwell found a set of partial differential equations which were so powerful that they could be used to derive the entire theory of electricity and magnetism, yet so simple that they could be written on the back of a postcard. Maxwell s equations are formulated in terms of divergence and curl. 5
To utilize Maxwell s equations in applications requires the divergence theorem and tokes s theorem. This is not the place to derive Maxwell s equations 1. However, we would like to mention that one can derive consequences from Maxwell s equations by means of the divergence theorem and tokes s theorem. Maxwell s equations are usually stated in terms of the charge density ρ(x, y, z, t), and the current density the electric field the magnetic field J(x, y, z, t), E(x, y, z, t), B(x, y, z, t). Maxwell s equations are a set of four partial differential equations involving the divergence and curl: E + B t = 0, E = ρ ɛ 0, B = 0, (3) B 1 c 2 E t = µ 0J. (4) Here c is the speed of light and ɛ 0 and µ 0 are constants. (The exact form of the equations depend on units used and other conventions.) In the case where all functions are independent of time, these equations simplify to E = ρ ɛ 0, B = 0, E = 0, B = µ 0 J. Often, the charge density ρ and the current density J are given, and one determines the electric and magnetic fields E and B by solving Maxwell s equations. The electric and magnetic fields can then determine the force acting on a test particle of charge Q in accordance with the formula F = QE + Q(v B), where v is the velocity of the particle. The divergence theorem applied to the Maxwell equation E = ρ ɛ 0 yields Gauss s law: If is a region in three-space bounded by a smooth surface, then E NdA = Edxdydz = 1 ρdxdydz = Q, ɛ 0 ɛ 0 1 A derivation of Maxwell s equations is presented in Lorrain and Corson, Electromagnetic fields and waves, econd equation, Freeman, an Francisco, 1970. 6
where Q denotes the total charge within. In other words, the total flux of E outward through the surface equals 1/ɛ 0 times the total charge within. On the other hand, tokes s theorem applied to the time-independent Maxwell equation B = µ 0 J yields Ampère s law: If is a surface bounded by a smooth curve, then B Tds = B NdA = µ 0 J NdA. In other words, the integral of the tangential component of B around the curve equals µ 0 times the total current flowing through. 7