FOCUS ON THEORY. We now consider a general change of variable, where x; y coordinates are related to s; t coordinates by the differentiable functions

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1 FOCUS ON HEOY 753 CHANGE OF VAIABLES IN A MULIPLE INEGAL In the previous sections, we used polar, clindrical, and spherical coordinates to simplif iterated integrals. In this section, we discuss more general changes of variable. In the process, we will see where the extra factor of r comes from when we change from Cartesian to polar coordinates and the factor ρ 2 sin f when we change from Cartesian to spherical coordinates. Polar Change of Variables evisited Consider the integral (x + ) da where is the region in the rst quadrant bounded b the circle x =16and the x and -axes. Writing the integral in Cartesian and polar coordinates we have 4 p 16 x 2 ß=2 4 (x + ) da = (x + ) d dx = (r cos + r sin )r drd : his is an integral over the rectangle in the r -space given b» r» 4,»» ß=2. he conversion from polar to Cartesian coordinates changes this rectangle into a quarter-disk. Figure 1 shows how a tpical rectangle (shaded) in the r -plane with sides of length r and corresponds to a curved rectangle in the x-plane with sides of length r and r. he extra r is needed because the correspondence between r; and x; not onl curves the lines r =1; 2; 3 :::into circles, it also stretches those lines around larger and larger circles. ß=2 r =2 r =3 3ß=8 3 = ß=4 ß=4 = ß=4 2 = ß=8 ß=8 = ß= r x Figure 1: A grid in the r -plane and the corresponding curved grid in the x-plane General Change of Variables We now consider a general change of variable, where x; coordinates are related to s; t coordinates b the dierentiable functions x = x(s; t) = (s; t): Just as a rectangular region in the r -plane corresponds to a circular region in the x-plane, a rectangular region,, in the st-plane corresponds to a curved region,, in the x-plane. We assume that the change of coordinates is one-to-one, that is, that each point corresponds to one point in.

2 754 CHAPE zero / FOCUS ON HEOY t (x(s; t + t);(s; t + t)) (s; t + t) (s; t) (s + s; t) ~ b ~a (x(s; t);(s; t)) (x(s + s; t);(s + s; t)) s x Figure 2: A small rectangle in the st-plane and the corresponding region of the x-plane We divide into small rectangles with sides of length s and t. (See Figure 2.) he corresponding piece of the x-plane is a quadrilateral with curved sides. If we choose s and t ver small, then b local linearit, is approximatel a parallelogram. ecall from Chapter 12 that the area of the parallelogram with sides ~a and ~ b is k~a ~ b k. hus, we need to nd the sides of as vectors. he side of corresponding to the bottom side of has endpoints (x(s; t);(s; t)) and (x(s + s; t);(s + s; t)), so in vector form that side is ~a =(x(s+ s; t) x(s; t))~i +((s+ s; t) (s; t))~j + ~ k ß Similarl, the side of corresponding to the left edge of is given b Computing the cross product, we get Area ßk~a ~ b kß ~ b ß t ~i + t ~j + ~ k: s = s t: Using determinant notation, we dene the Jacobian, t = =, as follows s ~i + s ~j + ~ k: t s : hus, we can write Area ß o compute f (x; ) da, where f is a continuous function, we look at the iemann sum obtained b dividing the region into the small curved regions, giving f (x; ) da ß f (x i ; j ) Area of ß f (x i ; j )

3 CHANGE OF VAIABLES IN A MULIPLE INEGAL 755 Each point (x i ; j ) corresponds to a point (s i ;t j ), so the sum can be written in terms of s and t: f (x(s i ;t j );(s i ;t j )) his is a iemann sum in terms of s and t,soas s and t approach, we get f (x; ) da = f (x(s; t);(s; t)) o convert an integral from x; to s; t coordinates we make three changes: 1. Substitute for x and in the integrand in terms of s and t. 2. Change the x region into an st region. 3. Introduce the absolute value of the Jacobian,, representing the change in the area element. Example 1 Verif that the Jacobian = r for polar coordinates x = r cos ; = r = cos sin r sin rcos = r cos2 + r sin 2 = r: Example 2 Find the area of the ellipse x2 a b 2 =1. Let x = as; = bt. hen the ellipse x 2 =a =b 2 =1in the x-plane corresponds to the circle s 2 + t 2 = 1 in the st-plane. he Jacobian is a = ab. hus, if we let be the ellipse in the b x-plane and the unit circle in the st-plane, we get Area of x-ellipse = 1 da = 1ab ds dt = ab ds dt = ab Area of st-circle = ßab: Change of Variables in riple Integrals For triple integrals, there is a similar formula. Suppose the dierentiable functions x = x(s; t; u); = (s; t; u); z = z(s; t; u) dene a change of variables from a region S in stu-space to a region W in xz-space. hen, the Jacobian of this change of variables is given b the ; t; u) = :

4 756 CHAPE zero / FOCUS ON HEOY Just as the Jacobian in two dimensions gives us the change in the area element, the Jacobian in three dimensions represents the change in the volume element. hus, we have W f (x; ; z) dx d dz = ; z) f (x(s; t; u);(s; t; u);z(s; t; u)) ds dt t; u) Problem 3 at the end of this section asks ou to verif that the Jacobian for the change of variables for spherical coordinates is ρ 2 sin f. he next example generalizes Example 2 to ellipsoids. Example 3 Find the volume of the ellipsoid x2 a b 2 + z2 c 2 =1. Let x = as; = bt; z = cu. he Jacobian is computed to be abc. he xz-ellipsoid corresponds to the stu-sphere s 2 + t 2 + u 2 =1. hus, as in Example 2, Volume of xz-ellipsoid = abc Volume of stu-sphere = abc 4 3 ß = 4 3 ßabc: Problems on Change of Variables in a Multiple Integral 1. Find the region in the x-plane corresponding to the region = f(s; t) j» s» 3;» t» 2g under the change of variables x = 2s 3t; = s 2t. Check that dx d = 2. Find the region in the x-plane corresponding to the region = f(s; t) j» s» 2; s» t» 2g under the change of variables x = s 2 ; = t. Check that dx d = 3. Compute the Jacobian for the change of variables into spherical coordinates: x = ρ sin f cos ; = ρ sin f sin ; z = ρ cos f: 4. For the change of variables x =3s 4t; =5s +2t, show that =1 5. Use the change of variables x =2s + t, = s t to compute the integral (x + ) da, where is the parallelogram formed b (; ), (3; 3), (5; 2), and (2; 1). 6. Use the change of variables x = 1 s, 2 = 1 3 t to compute the integral (x2 + 2 ) da, where is the region bounded b the curve 4x = Use the change of variables s = x, t = x 2 to compute x2 da, where is the region bounded b x =1; x =4; x 2 =1; x 2 =4. 8. Evaluate the integral and =. cos x x + dxd where is the triangle bounded b x + =1, x =,

5 CHANGE OF VAIABLES IN A MULIPLE INEGAL Find the area of the metal frames with one or four cutouts shown in Figure 3. Start with Cartesian coordinates x, aligned along one side. Consider slanted coordinates u = x, v = in which the frame is straightened. [Hint: First describe the shape of the cut-out in the uv-plane; second, calculate its area in the uv-plane; third, using Jacobians, calculate its area in the x-plane.] 135 f 135 f 45 f f Figure 3 3" 3" 3" 3" A river follows the path = f(x) where x, are in kilometers. Near the sea, it widens into a lagoon, then narrows again at its mouth. See Figure 4. At the point (x; ), the depth, d(x; ), of the lagoon is given b d(x; ) =4 16( f(x)) 2 4x 2 meters: he lagoon itself is described b d(x; ). What is the volume of the lagoon in cubic meters? [Hint: Use new coordinates u = x=2, v = f(x) and Jacobians.] Sea ( 1;f( 1)) I iver, = f(x) (1;f(1)) Lagoon Figure 4