4.4 Change of Variable in Integrals: The Jacobian

Transcription

1 4.4. CHANGE OF VAIABLE IN INTEGALS: THE JACOBIAN Change of Variable in Integrals: The Jacobian In this section, we generalize to multiple integrals the substitution technique used with definite integrals. For functions of two or more variables, there is a similar process we can use. It is a little bit more involved though. In addition, in higher dimensions, a change of variable can also be used to simplify the region of integration. We begin with an important definition The Jacobian The Jacobian, named after the German mathematician Carl Gustav Jacobi 84-85), plays an important role in higher dimensional mathematics. In this section, we see how it is used when changing variables to simplify a region of integration or an integrand. We begin with its definition. Definition 6 -D case) Suppose that x and y are two independent variables which can be expressed in term of two other independent variables u and v by the formula x g u, v) and y h u, v). The Jacobian of x and y with respect to u and v, denoted or J u, v), is u, v) J u, v) u, v) x u y u x v y v x y u v y u Definition 64 -D case) Suppose that x, y and z are three independent variables which can be expressed in term of three other independent variables u, v and w by the formula x g u, v, w), y h u, v, w) and z l u, v, w). x, y, z) The Jacobian of x, y and z with respect to u, v and w, denoted u, v, w) or J u, v), is x x x x, y, z) J u, v, w) u, v, w) u v w y y y u v w z z z u v w x x x x x Definition 65 The matrices u v u v w y y and y y y u v w u v z z z which u v w appear in the above definitions are called Jacobian matrices. Hence, the Jacobian is the determinant of the Jacobian matrix. x v

2 4 CHAPTE 4. MULTIPLE INTEGALS Example 66 ecall, when switching from Cartesian to polar coordinates, we have x r cos θ and y r sin θ. The Jacobian of x and y with respect to r and θ is r, θ) r, θ) x x r θ y y r θ cos θ r sin θ sin θ r cos θ r cos θ + r sin θ r Example 67 ecall, when switching from Cartesian to Spherical coordinates, we have x ρ sin φ cos θ, y ρ sin φ sin θ, and z ρ cos φ. The Jacobian of x, y, and z with respect to ρ, θ and φ is x x x x, y, z) ρ, θ, φ) ρ y ρ z ρ θ y θ z θ φ y φ z φ sin φ cos θ ρ sin φ sin θ ρ cos φ cos θ sin φ sin θ ρ sin φ cos θ ρ cos φ sin θ cos φ ρ sin φ The easiest way to compute this determinant is to expand it using the third row since one of its entries is. We obtain x, y, z) cos φ ρ sin φ sin θ ρ cos φ cos θ ρ, θ, φ) ρ sin φ cos θ ρ cos φ sin θ ρ sin φ sin φ cos θ ρ sin φ sin θ sin φ sin θ ρ sin φ cos θ cos φ ρ sin φ cos φ sin θ ρ sin φ cos φ cos θ ) ρ sin φ ρ sin φ cos θ + ρ sin φ sin θ ) ρ sin φ cos φ ρ sin φ ρ sin φ 4.4. Change of Variable You will recall in one-dimensional calculus, when given an integral of the form g u) f g u)) du, we performed the change of variable x g u) which gave us dx g u) du and thus g u) f g u)) du f x) dx

3 4.4. CHANGE OF VAIABLE IN INTEGALS: THE JACOBIAN 4 We can write this slightly differently as follows. Since x g u), dx du g u) hence, we have f x) dx f g u)) dx du du The Jacobian is what generalizes dx in the above formula. We begin with du the change of variable theorem for double integrals. We then look at several examples to see how one can benefit from a change of variable. These benefits include using a change of variable to simplify an integrand, using a change of variable to simplify a region of integration. As an application, we will look at double integrals in polar coordinates. Note that most of the explanations given below will be for regions in the xy-plane or for functions of two variables. General Case Let us first introduce some notation. Let denote a region in the xy-plane and S a region in the uv-plane. A change of variable is usually described as a transformation T from the uv-plane to the xy-plane given by T u, v) x, y) where x and y are given by x g u, v) y h u, v) We usually assume that the first order partials of g and h are continuous. When it is the case we say that T is a C transformation. Such a transformation will map a region S in the uv-plane into another region into the xy-plane see figure 4.4.). In most cases, we are given and are looking for a region S and a transformation T from S to in which S is simpler than. We will only consider simple cases here. When we talk about the Jacobian of the transformation T, we mean the Jacobian of the change of variable x g u, v) y h u, v) We begin with the change of variable theorem for integrals, given without proof. Theorem 68 Change of Variable in Double Integrals) Let and S be regions of the xy-planes and uv-planes respectively. Let T : S be a C transformation such that T u, v) x, y) where x g u, v) y h u, v)

4 44 CHAPTE 4. MULTIPLE INTEGALS such that each point in is the image of a unique point in S. If f is continuous on and on then u, v) f x, y) dxdy f g u, v), h u, v)) u, v) dudv S emark 69 It is important to understand that is the given region and the integral in terms of x and y is the given integral. We must find an appropriate change of variable in which the integral on the right in terms of u and v) is simpler. As stated, the theorem is deceiving. It makes it look like the integral on the left is easier than the integral on the right which appears to contain much more. But this is not the case in practice. emark 7 As in the one dimensional case, it is important to understand that when one performs a change of variable, not only the integrand changes, but also the region of integration. As mentioned already, there two possible reasons for performing a change of variable:. To obtain a simpler integrand.. To obtain an easier region over which to integrate. The ideal region is a rectangle with sides parallel to the coordinate axes. In this case, we can use Fubini s theorem. emark 7 Deciding what region works best for an integral takes practice. Finding the transformation T which maps one region into another is also very involved and could be the purpose of an entire course. Here, we will only consider simple examples for which the change of variable will be suggested.

5 4.4. CHANGE OF VAIABLE IN INTEGALS: THE JACOBIAN 45 Figure 4.: egion bounded by y, y 4, y x and y x Example 7 Evaluate 4 u x y and v y. y + y x y dxdy by applying the transformation The region corresponding to this integral is { x, y) : y 4, y x y + }. This is a type II region. It is shown in figure 4.. In the equations defining u and v, we need to solve for x and y since we need to compute the Jacobian of the transformation giving us xand y in terms of u and v. Simple algebra gives x u + v and y v. Therefore u, v) We also need to find what the new region, we call S will be. It is enough to find its boundaries. We illustrate how to do this in the table below: xy equations for the boundary of uv equations for the boundary of S Simplified uv equations x y u + v v v u x y + u + v v + v + u y v v y 4 v 4 v

6 46 CHAPTE 4. MULTIPLE INTEGALS Figure 4.: egion bounded by u, u, v and v. Hence S {u, v) : u, v }. This region is shown in figure 4.. We are now ready to apply the change of variable formula. 4 y + y x y dxdy u u, v) dudv ududv [ u ] dv dv x Example 7 Evaluate x + y y x) dydx using the transformation u x + y and v y x. First, let us find the region over which we are integrating. It is {x, y) : x, y x}. It is shown in figure 4.. First, we need to solve for x and y. Simple algebra

7 4.4. CHANGE OF VAIABLE IN INTEGALS: THE JACOBIAN 47 Figure 4.: egion bounded by x, y, x + y. suggests that x u v and y u + v. Next, we compute u, v) u, v). Finally, we need to find the corresponding uv region, we call it S. As in the previous example, we use a table. Not that this time the region is a triangle, hence its boundary consists of three lines. xy equations for the boundary of uv equations for the boundary of S Simplified uv equations u v x v u u + v y v u u v x + y + u + v u So, we see that S is the region bounded by u, v u and v u. It is shown

8 48 CHAPTE 4. MULTIPLE INTEGALS Figure 4.4: egion bounded by u, u v and v u. in figure 4.4.We can write it as a type I region. S {x, y) : u, u v u}. We are now ready to do the change of variable. x u x + y y x) dydx uv u u, v) dvdu [ ] v u u du u ) u u 7 du 9 9 u 9 9 u 7 du Application of Change of Variable to Polar Coordinates You will recall that the change of variable from Cartesian to polar coordinates is x r cos θ and y r sin θ. Also, from example 66, the Jacobian of this

9 4.4. CHANGE OF VAIABLE IN INTEGALS: THE JACOBIAN 49 transformation is we have r, θ) f x, y) dxdy r. Therefore, from the change of variable theorem, f r cos θ, r sin θ) r, θ) drdθ S f r cos θ, r sin θ) rdrdθ S where S is the region in the rθ-plane corresponding to in the xy-plane. Example 74 Evaluate e x +y dxdy where is the upper half portion of the unit circle. The region can be written as { x, y) : x and y x }. The corresponding region S is S {r, θ) : r and θ π}. Therefore e x +y dxdy re r drdθ S π re r drdθ We can evaluate the inner integral using the substitution u r hence du rdr thus re r dr e u du e ) You will note that in the process, we updated the limits of integration. It does not sow because they end up being the same. But make sure not to forget to do it. The original limits were and, they were limits for r. When we substitute and use u instead, we must find the limits on u. Since u r when r, u and when r, u. We can now finish the integral. π e x +y dxdy e ) dθ π e ) Example 75 Evaluate x + y ) da where is the annular region between the two circles x + y and x + y 5. The corresponding region S is { S r, θ) r } 5 and θ π

10 5 CHAPTE 4. MULTIPLE INTEGALS Therefore x + y ) da π 5 π 5 π π r 4 r cos θ + r sin θ ) rdrdθ r cos θ + r sin θ ) drdθ ) 4 cos θ + r 5 sin θ 6 cos θ sin θ dθ ) dθ Using the fact that cos + cos x θ, we have x + y ) ) π da + cos θ sin θ dθ 6π 4.4. Problems { u x y. Solve the system v x + y { u x + y. Solve the system v x y for x and y then find for x and y then find u, v). u, v).. Evaluate x xy y ) dxdy where is the region in the first quadrant bounded by the lines y x+4, y x+7, y x and y x+ by using the transformation u x y and v x + y. 4. Evaluate y and v x y. y x + ) e y x dxdy using the transformation u x + y 5. ewrite the given integrals from Cartesian to polar coordinates, then evaluate the integral. a) x dydx b) x a x x + y ) dydx c) a a dydx a x d) 6 y xdxdy

11 4.4. CHANGE OF VAIABLE IN INTEGALS: THE JACOBIAN Answers [. Solution is x u + v, y u + ] v. Therefore u, v). Solution is: [x u + v, y u v ]. Therefore u, v). Evaluate x xy y ) dxdy where is the region in the first quadrant bounded by the lines y x+4, y x+7, y x and y x+ by using the transformation u x y and v x + y. x xy y ) dxdy 4 4. Evaluate y and v x y. y x + ) e y x dxdy using the transformation u x + y y y x + ) e y x dxdy e + 5. ewrite the given integrals from Cartesian to polar coordinates, then evaluate the integral. a) x dydx π rdrdθ π b) c) a a d) 6 x a x x + y ) dydx π dydx π a x y xdxdy π π 4 a r drdθ π 8 rdrdθ πa 6 sin θ r cos θdrdθ 6