# 18.02 Multivariable Calculus Fall 2007

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1 MIT OpenCourseWare Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit:

2 V9. Surface Integrals Surface integrals are a natural generalization of line integrals: instead of integrating over a curve, we integrate over a surface in 3-space. Such integrals are important in any of the subjects that deal with continuous media (solids, fluids, gases), as well as subjects that deal with force fields, like electromagnetic or gravitational fields. Though most of our work will be spent seeing how surface integrals can be calculated and what they are used for, we first want to indicate briefly how they are defined. The surface integral of the (continuous) function f (x, y, z) over the surface S is denoted by You can think of ds as the area of an infinitesimal piece of the surface S. To define the integral (I), we subdivide the surface S into small pieces having area ASi, pick a point (xi, yi, zi) in the i-th piece, and form the Riemann sum As the subdivision of S gets finer and finer, the corresponding sums (2) approach a limit which does not depend on the choice of the points or how the surface was subdivided. The surface integral (1) is defined to be this limit. (The surface has to be smooth and not infinite in extent, and the subdivisions have to be made reasonably, otherwise the limit may not exist, or it may not be unique.) 1. The surface integral for flux. The most important type of surface integral is the one which calculates the flux of a vector field across S. Earlier, we calculated the flux of a plane vector field F(x, y) across a directed curve in the xy-plane. What we are doing now is the analog of this in space. We assume that S is oriented: this means that S has two sides and one of them has been designated to be the positive side. At each point of S there are two unit normal vectors, pointing in opposite directions; the positively directed unit normal vector, denoted by n, is the one standing with its base (i.e., tail) on the positive side. If S is a closed surface, like a sphere or cube - that is, a surface with no boundaries, so that it completely encloses a portion of 3-space -then by convention it is oriented so that the outer side is the positive one, i.e., so that n always points towards the outside of S. Let F(x, y, z) be a continuous vector field in space, and S an oriented surface. We define (3) flux of F through S = the two integrals are the same, but the second is written using the common and suggestive abbreviation ds = nds. If F represents the velocity field for the flow of an incompressible fluid of density 1, then F.n represents the component of the velocity in the positive perpendicular direction to the surface, and F. nds represents the flow rate across the little infinitesimal piece of surface

3 2 V. VECTOR INTEGRAL CALCLUS having area ds. The integral in (3) adds up these flows across the pieces of surface, so that we may interpret (3) as saying (4) flux of F through S = net flow rate across S, where we count flow in the direction of n as positive, flow in the opposite direction as negative. More generally, if the fluid has varying density, then the right side of (4) is the net mass transport rate of fluid across S (per unit area, per time unit). If F is a force field, then nothing is physically flowing, and one just uses the term "flux" to denote the surface integral, as in (3). We now show how to calculate the flux integral. It takes a few steps, and is best done by examples. The basic things are finding n and ds. We will begin with two surfaces where these are easy to calculate - the cylinder and the sphere. Then we will consider a general surface. Example 1. Find the flux of F = z i + x j + y k outward through the portion of the cylinder x2 + y2 = a2 in the first octant and below the plane z = h. Solution. The piece of cylinder is pictured. The word "outward" suggests that we orient the cylinder so that n points outward, i.e., away from the z- axis. Since by inspection n is radially outward and horizontal, xi +yj (5) n = a (This is the outward normal to the circle x2 +y2 = a2 in the xy-plane; n has no z-component since it is horizontal. We divide by a to make its length 1.) To get ds, the infinitesimal element of surface area, we use cylindrical coordinates to parametrize the cylinder: As the parameters 8 and z vary, the whole cylinder is traced out ; the piece we want satisfies ~ /2,0 5 z 5 h. The natural way to subdivide the cylinder is to use little pieces of curved rectangle like the one shown, bounded by two horizontal circles and two vertical lines on the surface. Its area ds is the product of its height and width: Having obtained n and ds, the rest of the work is routine. We express the integrand of our surface integral (3) in terms of z and 8: = (az cos 8 + a2 sin 8 cos 8) dz do, using (6). This last step is essential, since the dz and d8 tell us the surface integral will be calculated in terms of z and 8, and therefore the integrand must use these variables also. We can now calculate the flux through S: ah2 inner integral = -cos 8 + a2 h sin 8 cos 8 2 s i 8 1 ~ outer integral = sin 8 + a2h- = -(a + h).

4 V9. SURFACE INTEGRALS 3 Example 2. Find the flux of F = xz i + yz j + z2k outward through that part of the sphere x2 + y2 + z2 = a2 lying in the first octant (x, y, z, 2 0). Solution. Once again, we begin by finding n and ds for the sphere. We take the outside of the sphere as the positive side, so n points radially outward from the origin; we see by inspection therefore that where we have divided by a to make n a unit vector. To do the integration, we use spherical coordinates p, q5, 8. On the surface of the sphere, p = a, so the coordinates are just the two angles q5 and 8. The area element ds is most easily found using the volume element: dv = p2 sin q5 dp dq5 d8 = ds. dp = area. thickness so that dividing by the thickness dp and setting p = a, we get (9) ds = a2 sinq5dq5d8. Finally since the area element ds is expressed in terms of q5 and 8, the integration will be done using these variables, which means we need to express x, y, z in terms of q5 and 8. We use the formulas expressing Cartesian in terms of spherical coordinates (setting p = a since (x, y, z) is on the sphere): We can now calculate the flux integral (3). By (8) and (9), the integrand is Using (lo), and noting that x2 + y2 + z2 = a2, the integral becomes JLF-~~s = a4 l1i2 lri2 cos q5 sin q5 dq5 db

5 4 V. VECTOR INTEGRAL CALCLUS 2. Flux through general surfaces. For a general surface, we will use xyz-coordinates. Two forms for the equation of the surface, and the corresponding form for n are: VF n=f- (choose the right sign); l VFl -fzi - fgj + k n= (if n points "up") (f: fy2 1)lI2 + + The expression (11) for n uses the fact that the gradient V F is normal to the surface F = c; we divided by lvfl to make n a unit vector. To get the second form (ll'), write the surface in the form z - f (x, y) = 0, and then use (ll), taking F(x, y, z) to be z - f (x, y). As an example of the use of (ll), for the sphere x2 + y2 + z2 = a2, it gives immediately the unit normal vector 1 n = -(xi + yj +zk). a Most often however it is (11') that is used. We will need therefore the expression for ds in xy-coordinates. A natural choice for the infinitesimal element of surface is the infinitesimal piece of S lying over the area element dx dy in the xy-plane. The area of these two pieces are related by the basic formula dx dy (12) ds = -. In. kl The argument for (12) is as follows. Since ds is small, we can suppose to a rjk. first-order approximation that it lies in a plane P. Then the formula (12) is just a special case of a useful result relating the area of region R in the xy-plane with the area of the region R' lying above it in the plane P, namely: area R (13) area R' = - (n = unit normal to P). In. kl We can see that (13) is true in two steps: Step 1: It is true if R is a rectangle whose sides Au and Av are parallel and perpendicular to the line of intersection L. For as the picture shows, the sides of R' are Au and %,where y is both the dihedral angle between P and the xy-plane, and the angle between their respective normals n and k. Therefore which proves Step 1. Av Au Av area R' = Au.---- cosy In. k)' Step 2: To prove (13) for general plane areas R and R', we subdivide R into small rectangles AR of the type used in Step 1. This gives a corresponding subdivision of R' into rectangles AR'. (There will be left-over pieces, of course, but their area will be negligible if the subdivision is fine enough.)

6 V9. SURFACE INTEGRALS 5 Since (13) is true for each of the little pieces above and below, it is true for their corresponding sums as well, and these two sums approximate the area of R and R' arbitrarily closely. This completes the argument for (13), and therefore also for (12). If we apply (11') to (12), we get the explicit expression One can think of this as the two-dimensional analog of the one-dimensional formula for the infinitesimal arclength along a curve y = f (x): Actually, for calculating the surface integral representing flux, what is needed is not ds alone, but rather the combination nds. If we combine (11') and (129, the square roots miraculously cancel each other out, and we end up with the much simpler-looking vector formula (13) ds = n ds = (- f, i -f, j + k ) dx dy (n points "up") and this is the formula you will use most frequently in calculating surface integrals for the flux through a general surface given by z = f (x, y). Example 3. The portion of the plane 22-2y + z = 1 lying in the first octant forms a triangle S. Find the flux of F = xi + y j + z k through S; take the positive side of S as the one where the normal points "up". Solution. Writing the plane in the form z = y, we get by (13), where R is the region in the xy-plane over which S lies. (Note that since the integration is to be in terms of x and y, we had to express z in terms of x and y for this last step.) To see what R is explicitly, the plane intersects the three coordinate axes respectively at x = 112, y = -112, z = 1. So R is the region pictured; our integral has integrand 1,so its value is the area of R, which is 118. Remark. When we write z = f (x, y), we are agreeing to parametrize our surface using x and y as parameters. Thus the flux integral will be reduced to a double integral over a region R in the xy-plane, involving only x and y. Therefore you must get rid of z by using the relation z = f (x, y) after you have calculated the flux integral using (13). Then determine R (the projection of S onto the xy-plane), and supply the limits for the iterated integral over R.

7 6 V. VECTOR INTEGRAL CALCLUS 3. Other types of surface integrals. The general surface integral that we introduced at the beginning of this section can be used for many things other than calculating flux. Here are some examples. a) Surface area. We let the function f (x, y, z) = 1. Then the area of S = JL ds b) Mass, moments, charge etc. If S is a thin shell of material, of uniform thickness, and with density (in gms/unit area) given by 6(x, y, z), then mass of S = Jjs 6(x,l,2) ds, x-component of center of mass = T = -R x.6ds mass S with the y- and z-components of the center of mass defined similarly. If 6(x, y, z) represents an electric charge density, then the surface integral on the left above will give the total charge on S. And so on. c) Average value. The average value of a function f (x, y, z) over the surface S can be calculated by a surface integral: average value of f on s = - f(x, Y, z) d s. area S JJs In evaluating such integrals as (14), if the surface is a sphere or cylinder, then ds should be written down using the formulas in section 2 of V9, while if the surface is a more general one, given by z = f (x, y), then one uses the results of section 3 instead - unfortunately, the radical which appears in ds (see (12') will not disappear, since there is no n to cancel it. Example 4. Find the average distance along the earth of the points in the northern hemisphere from the North Pole. (Assume the earth is a sphere of radius a.) Solution. - We use (15) and spherical coordinates, choosing the coordinates so the North Pole is at z = a on the z-axis. The distance of the point (a,), 19) from (a, 0,O) is a), measured along the great circle, i.e., the longitude line - see the picture). We want to find the average of this function JL over the upper hemisphere 12= lti2 S. Integrating, and using (9), we get a) ds = a)a2 sin )d) di9 = 2ra3 ) sin )d) = 2ra3. lti2 (The last integral used integration by parts.) Since the area of S = 27ra2, we get using (15) the striking answer: average distance = a. Exercises: Section 6B

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