CHANGE OF VARIABLES - Multiple Integrals - Advanced Calculus of Several Variables

Advanced Calculus of Several Variables (1973)

Part IV. Multiple Integrals

Chapter 5. CHANGE OF VARIABLES

The student has undoubtedly seen change of variables formulas such as

and

which result from changes from rectangular coordinates to polar and spherical coordinates respectively. The appearance of the factor r in the first formula and the factor ρ² sin φ in the second one is sometimes “explained” by mythical pictures, such as Figure 4.24, in which it is alleged that dA is an “infinitesimal”

Figure 4.24

rectangle with sides dr and r dθ, and therefore has area r dr dθ. In this section we shall give a mathemtaically acceptable explanation of the origin of such factors in the transformation of multiple integrals from one coordinate system to another.

This will entail a thorough discussion of the technique of substitution for multivariable integrals. The basic problem is as follows. Let T : ⁿ → ⁿ be a mapping which is (continuously differentiable) and also -invertible on some neighborhood U of the set , meaning that T is one-to-one on U and that T⁻¹ : T(U) → U is also a mapping. Given an integrable function f : T(A) → , we would like to “transform” the integral ∫_T_(A) f into an appropriate integral over A (which may be easier to compute).

For example, suppose we wish to compute ∫_Q f, where Q is an annular sector in the plane ², as pictured in Fig. 4.25. Let T : ² → ² be the “polar coordinates” mapping defined by T(r, θ) = (r cos θ, r sin θ). Then Q = T(A), where A is the rectangle . If ∫_Q f can be transformed into an integral over A then, because A is a rectangle, the latter integral will be amenable to computation by iterated integrals (Fubini's theorem).

Figure 4.25

The principal idea in this discussion will be the local approximation of the transformation T by its differential linear mapping dT. Recall that dT_a : ⁿ → ⁿ is that linear mapping which “best approximates” the mapping h → T(a + h) − T(a) in a neighborhood of the point , and that the matrix of dT_a is the derivative matrix T′(a) = (D_jT_i(a)), where T₁, . . . , T_n are the component functions of T : ⁿ → ⁿ.

We therefore discuss first the behavior of volumes under linear distortions. That is, given a linear mapping λ : ⁿ → ⁿ and a contented set , we investigate the relationship between v(A) and v(λ(A)) [assuming, as we will prove, that the image λ(A) is also contented]. If λ is represented by the n × n matrix A, that is, λ(x) = Ax, we write

Theorem 5.1 If λ : ⁿ → ⁿ is a linear mapping, and is contented, then λ(B) is also contented, and

PROOFFirst suppose that det λ = 0, so the column vectors of the matrix of λ are linearly dependent (see by Theorem I.6.1). Since these column vectors are the images under λ of the standard unit vectors e₁, . . . , e_n in ⁿ, it then follows that λ(ⁿ) is a proper subspace of ⁿ. Since it is easily verified that any bounded subset of a proper subspace of ⁿ is negligible (Exercise 5.1), we see that (1) is trivially satisfied if det λ = 0.

If det λ ≠ 0, then a standard theorem of linear algebra asserts that the matrix A of λ can be written as a product

where each A_i is either of the form

or of the form

Hence λ is a composition λ = λ₁ λ₂ · · · λ_k, where λ_i is the linear mapping represented by the matrix A_i.

If A_i is of the form (2), with a appearing in the pth row and column, then

so it is clear that, if I is an interval, then λ_i(I) is also an interval, with v(λ_i(I)) = av(I). From this, and the definition of volume in terms of intervals, it follows easily that, if B is a contented set, then λ_i(B) is contented with v(λ_i(B)) = av(B) = det λ_i v(B).

We shall show below that, if A_i is of the form (3), then v(λ_i(B)) = v(B) = det λ_iv(B) for any contented set. Granting this, the theorem follows, because we then have

as desired, since the determinant of a product of matrices is the product of their determinants.

So it remains only to verify that λ_i preserves volumes if the matrix A_i of λ_i is of the form (3). We consider the typical case in which the off-diagonal element of A_i is in the first row and second column, that is,

We show first that λ_i preserves the volume of an interval

If we write I = I′ × I″ where and , then (4) shows that

where I^* is the pictured parallelogram in ² (Fig. 4.26). Since it is clear that v(I^*) = v(I′), Exercise 4.1 gives

Figure 4.26

Now let B be a contented set in ⁿ, and, given ε > 0, consider nonoverlapping intervals I₁, . . . , I_p and intervals J₁, . . . , J_q such that

and

Since we have seen that λ_i preserves the volumes of intervals, we then have

and

Finally it follows easily from these inequalities that λ_i(B) is contented, with v(λ_i(B)) = v(B) as desired (see Exercise 5.2).

We are now in a position to establish the invariance of volume under rigid motions.

Corollary 5.2 If is a contented set and ρ : ⁿ → ⁿ is a rigid motion, then ρ(A) is contented and v(ρ(A)) = v(A).

The statement that ρ is a rigid motion means that ρ = τ_a μ, where τ_a is the translation by a, τ_a(x) = x + a, and μ is an orthogonal transformation, so that det μ = 1 (see Exercise 1.6.10). The fact that ρ(A) is contented with v(ρ(A)) = v(A) therefore follows immediately from Theorem 5.1.

Example 1 Consider the ball

of radius r in ⁿ. Note that B_rⁿ is the image of the unit n-dimensional ball under the linear mapping T(x) = rx. Since det T′ = det T = rⁿ, Theorem 5.1 gives

Thus the volume of an n-dimensional ball is proportional to the nth power of its radius. Let α_n denote the volume of the unit n-dimensional ball B₁ⁿ, so

In Exercises 5.17, 5.18, and 5.19, we shall see that

These formulas are established by induction on m, starting with the familiar α₂ = π and α₃ = 4π/3.

In addition to the effect on volumes of linear distortions, discussed above, a key ingredient in the derivation of the change of variables formula is the fact that, if F : ⁿ → ⁿ is a mapping such that dF₀ = I (the identity transformation of ⁿ), then in a sufficiently small neighborhood of 0, F differs very little from the identity mapping, and in particular does not significantly alter the volumes of sets within this neighborhood.

This is the effect of the following lemma, which was used in the proof of the inverse function theorem in Chapter III. Recall that the norm λ of the linear mapping λ : ⁿ → ⁿ is by definition the maximum value of λ(x)₀ for all , where

and . Thus C_r is a “cube” centered at 0 with (r, r, . . . , r) as one vertex, and is referred to as “the cube of radius r centered at the origin.”

The lemma then asserts that, if dF_x differs (in norm) only slightly from the identity mapping I for all , then the image F(C_r) contains a cube slightly smaller than C_r, and is contained in one slightly larger than C_r.

Lemma 5.3 (Lemma III.3.2) Let U be an open set in ⁿ containing C_r, and F : U → ⁿ a mapping such that F(0) = 0 and dF₀ = I. Suppose also that there exists such that

for all . Then

(see Fig. 4.27).

Now let Q be an interval centered at the point , and suppose that T : U → ⁿ is a -invertible mapping on a neighborhood U of Q. Suppose in addition that the differential dT_x is almost constant on Q, dT_x ≈ dT_a, and in particular that

Figure 4.27

for all x Q. Also let Q_1−ε and Q_1+ε be intervals centered at a and both “similar” to Q, with each edge of Q_1±ε equal (in length) to 1 ± ε times the corresponding edge of Q, so v(Q_1±ε) = (1 ± ε)ⁿv(Q). (see Fig. 4.28.)

Then the idea of the proof of Theorem 5.4 below is to use Lemma 5.3 to show that (6) implies that the image set T(Q) closely approximates the “parallelepiped” dT_a(Q), in the sense that

It will then follow from Theorem 5.1 that

Figure 4.28

Theorem 5.4 Let Q be an interval centered at the point , and suppose T : U → ⁿ is a -invertible mapping on a neighborhood U of Q. If there exists such that

for all x Q, then T(Q) is contented with

PROOFWe leave to Exercise 5.3 the proof that T(Q) is contented, and proceed to verify (7).

First suppose that Q is a cube of radius r. Let τ_a denote translation by a, τ_a(x) = a + x, so τ_a(C_r) = Q. Let b = T(a), and λ = dT_a. Let F = λ⁻¹ τ_b⁻¹ T τ_a (see Fig. 4.29).

By the chain rule we obtain

Since the differential of a translation is the identity mapping, this gives

for all . Consequently hypothesis (6) implies that F : C_r → ⁿ satisfies (5). Consequently Lemma 5.3 applies to yield

Figure 4.29

Since , Theorem 5.1 implies that

Since v(Q) = v(C_r), we therefore obtain (7) upon multiplication of (8) by det T′(a). This completes the proof in case Q is a cube.

If Q is not a cube, let ρ : ⁿ → ⁿ be a linear mapping such that ρ(C₁) is congruent to Q, and τ_a ρ(C₁) = Q in particular. Then v(Q) = det ρ v(C₁) by Theorem 5.1. Let

Then the chain rule gives

because dρ = ρ since ρ is linear. Therefore

by (6), so S : C₁ → ⁿ itself satisfies the hypothesis of the theorem, with C₁ a cube.

What we have already proved therefore gives

Since S(C₁) = T(Q) and

this last inequality is (7) as desired.

Since (1 ± ε)ⁿ ≈ 1 if ε is very small, the intuitive content of Theorem 5.4 is that, if dT_x is approximately equal to dT_a for all x Q (and this should be true if Q is sufficiently small), then v(T(Q)) is approximately equal to det T′(a)v(Q). Thus the factor det T′(a) seems to play the role of a “local magnification factor.” With this interpretation of Theorem 5.4, we can now give a heuristic derivation of the change of variables formula for multiple integrals.

Suppose that T : ⁿ → ⁿ is -invertible on a neighborhood of the interval Q, and suppose that f : ⁿ → is an integrable function such that f T is also integrable.

Let = {Q₁, . . . , Q_k} be a partition of Q into very small subintervals, and = {a₁, . . . , a_k} the selection for such that a_i is the center-point of Q_i (Fig. 4.30). Then

The proof of the main theorem is a matter of supplying enough epsilonics to convert the above discussion into a precise argument.

Figure 4.30

Theorem 5.5Let Q be an interval in ⁿ, and T : ⁿ → ⁿ a mapping which is -invertible on a neighborhood of Q. If f : ⁿ → is an integrable function such that f T is also integrable, then

PROOFLet η > 0 be given to start with. Given a partition = {Q₁, . . . , Q_k} of Q, let = {a₁, . . . , a_k} be the selection consisting of the center-points of the intervals of . By Theorem 3.4 there exists δ₁ > 0 such that

if mesh < δ₁, where R denotes the Riemann sum

Choose A > 0 and M > 0 such that

for all x Q. This is possible because det T′(x) is continuous (hence bounded) on Q, and f T is integrable (hence bounded) on Q. If

then

where

By Theorem 3.4 there exists δ₂ > 0 such that, if mesh < δ₂, then any two Riemann sums for f T on differ by less than η/6A, because each differs from ∫_Qf T by less than η/12A. It follows that

because there are Riemann sums for f T arbitrarily close to both and . Hence (11) and (12) give

because each Δ_i < A. Summarizing our progress thus far, we have the Riemann sum R which differs from ∫_Q (f T) det T′ by less than η/2, and lies between the two numbers α and β, which in turn differ by less than η/6.

Next we want to locate ∫_T_(Q)f between two numbers α′ and β′ which are close to α and β, respectively. Since the sets T(Q_i) are contented (by Exercise 5.3) and intersect only in their boundaries,

by Proposition 2.7. Therefore

by Proposition 2.5. Our next task is to estimate α′ − α and β′ − β, and this is where Theorem 5.4 will come into play.

Choose such that

Since T is -invertible in a neighborhood of Q, we can choose an upper bound B for the value of the norm . By uniform continuity there exists δ₃ > 0 such that, if mesh < δ₃, then

for all . This in turn implies that

for all . Consequently Theorem 5.4 applies to show that v(T(Q_i)) lies between

as does Δ_i v(Q_i).

It follows that

if the mesh of is less than δ₃. We now fix the partition with mesh less than δ = min(δ₁, δ₂, δ₃). Then

by (15) and (16). Similarly

We are finally ready to move in for the kill. If α^* = min(α, α′) and β^* = max(β, β′) then, by (11) and (14), both R and ∫_T_(Q) f lie between α^* and β^*.

Since (13), (17), and (18) imply that β^* − α^* < η/2, it follows that

Thus ∫_T_(Q) f and ∫_Q (f T)det T′ both differ from R by less than η/2, so

This being true for every η > 0, we conclude that

Theorem 5.5 is not quite general enough for some of the standard applications of the change of variables formula. However, the needed generalizations are usually quite easy to come by, the real work having already been done in the proof of Theorem 5.5. For example, the following simple extension will suffice for most purposes.

Addendum 5.6 The conclusion of Theorem 5.5 still holds if the mapping T : ⁿ → ⁿ is only assumed to be -invertible on the interior of the interval Q (instead of on a neighborhood of Q).

PROOFLet Q^* be an interval interior to Q, and write K = Q − int Q^* (Fig. 4.31). Then Theorem 5.5 as stated gives

Figure 4.31

But

and

and the integrals over K and T(K) can be made as small as desired by making the volume of K, and hence that of T(K), sufficiently small. The easy details are left to the student (Exercise 5.4).

Example 2Let A denote the annular region in the plane bounded by the circles of radii a and b centered at the origin (Fig. 4.32). If T : ² → ² is the “polar coordinates mapping” defined by

Figure 4.32

(for clarity we use r and θ as coordinates in the domain, and x and y as coordinates in the image), then A is the image under T of the rectangle

T is not one-to-one on Q, because T(r, 0) = T(r, 2π), but is -invertible on the interior of Q, so Addendum 5.6 gives

Since det T′(r, θ) = r, Fubini's theorem then gives

Example 3Let A be an “ice cream cone” bounded above by the sphere of radius 1 and below by a cone with vertex angle π/6 (Fig. 4.33). If T: ³ → ³ is the “spherical coordinates mapping” defined by

then A is the image under T of the interval

Figure 4.33

T is not one-to-one on Q (for instance, the image under T of the back face ρ = 0 of Q is the origin), but is -invertible on the interior of Q, so Addendum 5.6 together with Fubini's theorem gives

since det T′(ρ, φ, θ) = ρ² sin φ.

Example 4To compute the volume of the unit ball , take f(x) ≡ 1, T the spherical coordinates mapping of Example 2, and

Then T(Q) = B³, and T is -invertible on the interior of Q. Therefore Addendum 5.6 yields

the familiar answer.

In integration problems that display an obvious cylindrical or spherical symmetry, the use of polar or spherical coordinates is clearly indicated. In general, in order to evaluate ∫_Rf by use of the change of variables formula, one chooses the transformation T (often in an ad hoc manner) so as to simplify either the function f or the region R (or both). The following two examples illustrate this.

Example 5Suppose we want to compute

where R is the region in the first quadrant which is bounded by the coordinate axes and the curve (a parabola). The substitution seems promising (it simplifies the integrand considerably), so we consider the mapping T from uv-space to xy-space defined by x = u², y = v² (see Fig. 4.34). It maps onto R the triangle Q in the uv-plane bounded by the co-

Figure 4.34

ordinate axes and the line u + v = 1. Now det T′ = 4uv, so the change of variables formula (which applies here, by Exercise 5.4) gives

This iterated integral (obtained from Theorem 4.3) is now easily evaluated using integration by substitution (we omit the details).

Example 6Consider ∫∫_R (x² + y²) dx dy, where R is the region in the first quadrant that is bounded by the hyperbolas xy = 1, xy = 3, x² − y² = 1, and x² − y² = 4. We make the substitution u = xy, v = x² − y² so as to simplify the region; this transformation S from xy-space to uv-space maps the region R onto the rectangle Q bounded by the lines u = 1, u = 3, v = 1, v = 4 (see Fig. 4.35). From u = xy, v = x² − y² we obtain

Figure 4.35

The last two equations imply that S is one-to-one on R; we are actually interested in its inverse T = S⁻¹. Since S T = I, the chain rule gives S′(T(u, v)) T′(u, v) = I, so

Consequently the change of variables formula gives

Multiple integrals are often used to compute mass. A body with mass consists of a contented set A and an integrable density function μ : A → ; its mass M(A) is then defined by

Example 7Let A be the ellipsoidal ball , with its density function μ being constant on concentric ellipsoids. That is, there exists a function g : [0, 1] → such that μ(x) = g(ρ) at each point x of the ellipsoid

In order to compute M(A), we intoduce ellipsoidal coordinates by

The transformation T defined by these equations maps onto A the interval

and is invertible on the interior of Q. Since det T′ = abcρ² sin φ, Addendum 5.6 gives

For instance, taking g(ρ) ≡ 1 (uniform unit density), we see that the volume of the ellipsoid is (as seen previously in Exercise 4.6).

A related application of multiple integrals is to the computation of force. A force field is a vector-valued function, whereas we have thus far integrated only scalar-valued functions. The function G : ⁿ → ^m is said to be integrable on if and only if each of its component functions g₁, . . . , g_m is, in which case we define

Thus vector-valued functions are simply integrated componentwise.

As an example, consider a body A with density function μ : A → , and let ξ be a point not in A. Motivated by Newton's law of gravitation, we define the gravitational force exerted by A, on a particle of mass m situated at ξ, to be the vector

where γ is the “gravitational constant.”

For example, suppose that A is a uniform rod (μ = constant) coincident with the unit interval [0, 1] on the x-axis, and ξ = (0, 1). (see Fig. 4.36.) Then the gravitational force exerted by the rod A on a unit mass at ξ is

For a more interesting example, see Exercises 5.13 and 5.14.

We close this section with a discussion of some of the classical notation and terminology associated with the change of variables formula. Given a differentiable mapping T : ⁿ → ⁿ, recall that the determinant det T′(a) of the deriva-

Figure 4.36

tive matrix T′(a) is called the Jacobian (determinant) of T at a. A standard notation (which we have used in Chapter III) for the Jacobian is

where T₁, . . . , T_n are as usual the component functions of T, thought of as a mapping from u-space (with coordinates u₁, . . . , u_n) to x-space (with coordinates x₁, . . . , x_n). If we replace the component functions T₁, . . . , T_n in the Jacobian by the variables x₁, . . . , x_n which they give,

then the change of variables formula takes the appealing form

With the abbreviations

and

the above formula takes the simple form

This form of the change of variables formula is particularly pleasant because it reads the same as the familar substitution rule for single-variable integrals (Theorem 1.7), taking into account the change of sign in a single-variable integral which results from the interchange of its limits of integration.

This observation leads to an interpretation of the change of variables formula as the result of a “mechanical” substitution procedure. Suppose T is a differentiable mapping from uv-space to xy-space. In the integral ∫_T_(Q) f(x, y) dx dy we want to make the substitutions

suggested by the chain rule. In formally multiplying together these two “differential forms,” we agree to the conventions

for no other reason than that they are necessary if we are to get the “right” answer. We then obtain

Note that, to within sign, this result agrees with formula (20), according to which dx dy should be replaced by ∂(x, y)/∂(u, v)du dv when the integral is transformed.

Thus the change of variables (x, y) = T(u, v) in a 2-dimensional integral ∫_T_(Q)f(x, y) dx dy may be carried out in the following way. First replace x and y in f(x, y) by their expressions in terms of u and v, then multiply out dx dy as above using (21) and (22), and finally replace the coefficient of du dv, which is obtained, by its absolute value,

For example, in a change from rectangular to polar coordinates, we have x = r cos θ and y = r sin θ, so

where the sign is correct because .

The relation dv du = −du dv requires a further comment. Once having agreed upon u as the first coordinate and v as the second one, we agree that

while

So, in writing multidimensional integrals with differential notation, the order of the differentials in the integrand makes a difference (of sign). This should be contrasted with the situation in regards to iterated integrals, such as

in which the order of the differentials indicates the order of integration. If parentheses are consistently used to indicate iterated integrals, no confusion between these two different situations should arise.

The substitution process for higher-dimensional integrals is similar. For example, the student will find it instructive to verify that, with the substitution

formal multiplication, using the relations

yields

(Exercise 5.20). These matters will be discussed more fully in Chapter V.

Exercises

5.1Let A be a bounded subset of ^k, with k < n. If f: ^k → ⁿ is a mapping, show that f(A) is negligible. Hint: Let Q be a cube in ^k containing A, and a partition of Q into N^k cubes with edgelength r/N, where r is the length of the edges of Q. Since f is on Q, there exists c such that for all . It follows that .

5.2Show that A is contented if and only if, given ε > 0, there exist contented sets A₁ and A₂ such that and v(A₂ − A₁) < ε.

5.3(a) If is negligible and T: ⁿ → ⁿ is a mapping, show that T(A) is negligible. Hint: Apply the fact that, given an interval Q containing A, there exists c > 0 such that T(x) − T(y) < cx − y for all .

(b) If A is a contented set in ⁿ and T: ⁿ → ⁿ is a mapping which is -invertible on the interior of A, show that T(A) is contented. Hint: Show that , and then apply (a).

5.4Establish the conclusion of Theorem 5.5 under the hypothesis that Q is a contented set such that T is -invertible on the interior of Q. Hint: Let Q₁, . . . , Q_k be nonoverlapping intervals interior to Q, and . Then

by Theorem 5.5. Use the hint for Exercise 5.3(a) to show that v(T(K)) can be made arbitrarily small by making v(K) sufficiently small.

5.5Consider the n-dimensional solid ellipsoid

Note that E is the image of the unit ball Bⁿ under the mapping T: ⁿ → ⁿ defined by

Apply Theorem 5.1 to show that v(E) = a₁ a₂ · · · a_nv(Bⁿ). For example, the volume of the 3-dimensional ellipsoid is .

5.6Let R be the solid torus in ³ obtained by revolving the circle , in the yz-plane, about the z-axis. Note that the mapping T: ³ → ³ defined by

maps the interval and onto this torus. Apply the change of variables formula to calculate its volume.

5.7Use the substitution u = x − y, v = x + y to evaluate

where R is the first quadrant region bounded by the coordinate axes and the line x + y = 1.

5.8Calculate the volume of the region in ³ which is above the xy-plane, under the paraboloid z = x² + y², and inside the elliptic cylinder x²/9 + y²/4 = 1. Use elliptical coordinates x = 3r cos θ, y = 2r sin θ.

5.9Consider the solid elliptical rod bounded by the xy-plane, the plane z = αx + βy + h through the point (0, 0, h) on the positive z-axis, and the elliptical cylinder x²/a² + y²/b² = 1. Show that its volume is πabh (independent of α and β). Hint: Use elliptical coordinates x = ar cos θ, y = br sin θ.

5.10Use “double polar” coordinates defined by the equations

to show that the volume of the unit ball is .

5.11Use the transformation defined by

to evaluate

where R is the region in the first quadrant of the xy-plane which is bounded by the circles x² + y² = 6x, x² + y² = 4x, x² + y² = 8y, x² + y² = 2y.

5.12Use the transformation T from rθt-space to xyz-space, defined by

to find the volume of the region R which lies between the paraboloids z = x² + y² and z = 4(x² + y²), and also between the planes z = 1 and z = 4.

5.13Consider a spherical ball of radius a centered at the origin, with a spherically symmetric density function d(x) = g(ρ).

(a)Use spherical coordinates to show that its mass is

(b)Show that the gravitational force exerted by this ball on a point mass m located at the point ξ = (0, 0, c), with c > a, is the same as if its mass were all concentrated at the origin, that is,

Hint:Because of symmetry considerations, you may assume that the force is directed toward 0, so only its z-component need be explicitly computed. In spherical coordinates (see Fig. 4.37),

Figure 4.37

Change the second variable of integration from φ to w, using the relations

so 2w dw = 2ρc sin φ dφ, and w cos α + ρ cos φ = c (why ?).

5.14Consider now a spherical shell, centered at the origin and defined by , with a spherically symmetric density function. Show that this shell exerts no gravitational force on a point mass m located at the point (0, 0, c) inside it (that is, c < a). Hint: The computation is the same as in the previous problem, except for the limits of integration on ρ and w.

5.15Volume of Revolution. Suppose g : [a, b] → is a positive-valued continuous function, and let

Denote by C the set generated by revolving A about the z-axis (Fig. 4.38); that is,

Figure 4.38

We wish to calculate v(C) = ∫_c 1. Noting that C is the image of the contented set

under the cylindrical coordinates mapping T : ³ → ³ defined by T(r, θ, z) = (r cos θ, r sin θ, z), show that the change of variable and repeated integrals theorems apply to give

5.16Let A be a contented set in the right half xz-plane x > 0. Define , the x-coordinate of the centroid of A, by = [1/v(A)] ∫∫_A x dx dz. If C is the set obtained by revolving A about the z-axis, that is,

then Pappus' theorem asserts that

that is, that the volume of C is the volume of A multiplied by the distance traveled by the centroid of A. Note that C is the image under the cylindrical coordinates map of the set