Partial Data Problems and Unique Continuation in Scalar and Vector Field Tomography

We prove that if P(D) is some constant coefficient partial differential operator and f is a scalar field such that P(D)f vanishes in a given open set, then the integrals of f over all lines intersecting that open set determine the scalar field uniquely everywhere. This is done by proving a unique continuation property of fractional Laplacians which implies uniqueness for the partial data problem. We also apply our results to partial data problems of vector fields.


Introduction
Let f be a scalar field and V ⊂ R n a nonempty open set where n ≥ 2. We study the following partial data problem in X-ray tomography: can we say something about f if we know the integrals of f over all lines intersecting V ? Especially, we are interested in the uniqueness problem which can be formulated in terms of the X-ray transform X 0 as follows: if X 0 f = 0 on all lines which intersect V , does it follow that f = 0? In general, the answer is "no" [30] and one has to put some conditions on f | V . We prove that if there is some constant coefficient partial differential operator P(D) such that Communicated by Eric Todd Quinto. P(D) f | V = 0 and X 0 f = 0 on all lines intersecting V , then f = 0. This generalizes a recent partial data result in [21]. As a special case we obtain that if f is for example polynomial or (poly)harmonic in V , then f is uniquely determined by its partial X-ray data.
The partial data result is proved by using a unique continuation property of fractional Laplacian (− ) s . We prove that if s ∈ (−n/2, ∞) \ Z and there is some constant coefficient partial differential operator P(D) such that P(D) f | V = (− ) s f | V = 0, then f = 0. This generalizes earlier results about unique continuation of fractional Laplacians [6,15]. The unique continuation of (− ) s implies a unique continuation result for the normal operator N 0 of the X-ray transform X 0 , and the uniqueness for the partial data problem follows directly from the unique continuation of N 0 . This approach which uses the unique continuation of the normal operator in proving uniqueness for partial data problems was also used in [6,21,22].
We also study partial data problems of vector fields. Let F be a vector field and denote by dF its exterior derivative or curl which components are (dF) i j = ∂ i F j − ∂ j F i . We prove that if there are some constant coefficient partial differential operators P i j (D) such that P i j (D)(dF) i j | V = 0 and the integrals of F over all lines intersecting V vanish, then F must be a potential field (F is the gradient of some scalar field). This is a generalization of a recent result in [22]. The partial data result is proved by using a relation between the normal operator of the X-ray transform of scalar fields and the normal operator of the X-ray transform of vector fields (see lemma 4.4). This allows one to reduce the partial data problem for the vector field F to partial data problems for the scalar fields (dF) i j . As a special case analogous to the scalar result, we obtain that if F is for example componentwise polynomial or (poly)harmonic in V , then the solenoidal part of F is uniquely determined by the partial X-ray data of F.
The partial data problems we study have a relation to the region of interest (ROI) tomography [4,24,25,30,47]. The main goal in such imaging problems is to determine the attenuation inside a small part of a human body (region of interest) by using only the X-ray data on lines which go through the ROI. This for example reduces the needed Xray dose which is given to the patient. Our results imply that if the attenuation f satisfies P(D) f | V = 0 for some open subset V of the ROI and some constant coefficient partial differential operator P(D), then f is uniquely determined by its partial X-ray data on lines which intersect the ROI. Note that f is uniquely determined not only in the ROI but also outside the ROI. Concrete examples of admissible functions are listed in Sect. 1.2 below. In general, f does not have to be smooth and it can have singularities in the ROI. We also note that our proof for uniqueness does not give stability for the partial data problem. Especially, outside the ROI we have invisible singularities which cannot be seen by the X-ray data and the reconstruction of such singularities is not stable (see remark 1.5 and [26, 35, 36]).
Similar ROI tomography problems can be studied in the case of vector fields. In vector field tomography the usual objective is to determine the velocity field of a fluid flow using acoustic travel time or Doppler backscattering measurements [31,32,40]. Assuming that the velocity of the fluid flow is much smaller than the speed of the propagating signal one can linearize the problem. Linearization then leads to the X-ray transform of the velocity field. Our results imply that if the velocity field F satisfies P i j (D)(dF) i j | V = 0 for some open subset V of the ROI and some constant coefficient partial differential operators P i j (D), then the solenoidal part of F is uniquely determined everywhere by the partial X-ray data of F on lines intersecting the ROI. The examples of admissible vector fields are the same as in the scalar case, only interpreted componentwise. As in the scalar case, F can have singularities in the ROI, and our proof does not give stability for the partial data problem (since it is based on reduction to the scalar case).
The article is organized as follows. In Sect. 1.1 we introduce our notation, in Sect. 1.2 we give examples of admissible functions, in Sect. 1.3 we give our main theorems and in Sect. 1.4 we discuss some related results. We go through the theory of distributions and the X-ray transform in Sect. 2, and study the space of admissible functions in Sect. 3. Finally, we prove our main results in Sect. 4.

Notation
We quickly go through the notation used in our main theorems. More detailed information about distributions and the X-ray transform of scalar and vector fields can be found in Sect. 2.
We denote by f a scalar field. The set O C (R n ) is the space of rapidly decreasing distributions and the space is the set of all continuous functions which decay faster than any polynomial at infinity. We let X 0 be the X-ray transform of scalar fields and it maps a function to its line integrals (see equations (8) and (10)). The normal operator is N 0 = X * 0 X 0 where X * 0 is the adjoint of X 0 (see equations (11) and (14)).
We denote by F a vector field. The notation F ∈ (E (R n )) n means that F = (F 1 , . . . , F n ) where F i ∈ E (R n ) for all i = 1, . . . , n. The exterior derivative of F is written in components as (dF) i j = ∂ i F j − ∂ j F i . For scalar fields φ the notation dφ denotes the gradient of φ. We let X 1 be the X-ray transform of vector fields which maps a vector field to its line integrals (see equations (16) and (17)). The normal operator is N 1 = X * 1 X 1 where X * 1 is the adjoint of X 1 (see equations (18) and (21)). We let H r (R n ) be the fractional L 2 -Sobolev space of order r ∈ R and H −∞ (R n ) = r ∈R H r (R n ). We define the fractional Laplacian as

Admissible Functions
We denote by P the set of all polynomials in R n with complex coefficients with the convention that the zero polynomial P ≡ 0 does not belong to P. A polynomial P ∈ P of degree m ∈ N induces a constant coefficient partial differential operator P(D) of order m ∈ N by setting P(D) = |α|≤m a α D α where a α ∈ C, D α = D α 1 1 · · · D α n n , D j = −i∂ j and α = (α 1 , . . . , α n ) ∈ N n is a multi-index such that |α| = α 1 +. . .+α n .

The set of admissible functions A V is defined as
where V ⊂ R n is some nonempty open set. Examples of admissible functions include If P(D) is a hypoelliptic operator, then the condition P(D) f | V = 0 already implies that f is smooth in V (see [19,29]). Basic examples of hypoelliptic operators are elliptic operators such as integer powers of Laplacians ((− ) k where k ∈ N) and also the non-elliptic heat operator ∂ t − . However, there are non-smooth distributions f | V which satisfy the condition P(D) f | V = 0 for some P ∈ P and therefore f can have singularities in V . For example, the wave operator ∂ 2 t − is not hypoelliptic and has non-smooth weak solutions. Another example of a non-hypoelliptic operator is the partial derivative ∂ x i : if f | V is independent of x i , then the behaviour with respect to the other variables can be singular.

Main Results
In this section we give our main theorems. The proofs of the results can be found in Sect. 4.
Our main theorem is the following unique continuation result for the fractional Laplacian. For compactly supported distributions we get a slightly stronger result.
From the unique continuation of fractional Laplacians we immediately obtain the following unique continuation result for the normal operator of the X-ray transform of scalar fields. The reason is that the normal operator can be written as N 0 = (− ) −1/2 up to a constant factor (see Sect. 2.2).
. In order to use theorem 1.1 in the case s = −1/2 and n ≥ 2, and to guarantee that The unique continuation of N 0 implies uniqueness for the following partial data problem.
The case where f is polynomial in V is previously known in two dimensions [24,47].
It is important to notice that from the vector space structure of admissible functions A V it follows that theorem 1.4 is indeed a uniqueness result: if f 1 and f 2 satisfy intersecting V , then f 1 = f 2 (see proposition 3.4 and remark 3.5 for more details). Especially, the equality of the X-ray data on all lines intersecting V implies that the scalar fields are equal everywhere even though f 1 and f 2 a priori can have very different behaviour in V since P 1 (D) can be different from P 2 (D).

Remark 1.5
Our proof for theorem 1.4 gives only uniqueness but not stability for the partial data problem. In theorem 1.4 we eventually have to assume that f is not supported in V since otherwise we would have P(D) f = 0 everywhere and therefore f = 0 without assuming anything about the X-ray data (see the proof of theorem 1.1). When f is supported outside V we do not have access to all singularities of f via the X-ray data, i.e. we have invisible singularities outside V . It is known that the recovery of such invisible singularities is not stable [26,35,36]. Remark 1.6 Similar results as in theorems 1.3 and 1.4 also hold for the d-plane transform R d when d is odd (see corollaries 1 and 2 on page 646 in [6]). The d-plane transform R d takes a scalar field and integrates it over d-dimensional affine planes where 0 < d < n. The case d = 1 corresponds to the X-ray transform and d = n − 1 to the Radon transform. The normal operator of the d-plane transform is the com- is the adjoint of R d and it can be expressed as N d = (− ) −d/2 up to a constant factor (see [6,17]). Hence N d admits the same unique continuation property as in theorem 1. From the unique continuation of fractional Laplacians we also obtain a partial data result for the X-ray transform of vector fields. The normal operators satisfy the relationship d(N 1 F) = N 0 (dF) up to a constant factor (see lemma 4.4). Hence the unique continuation and partial data problems of vector fields can be reduced to the corresponding problems for scalar fields, namely the components (dF) i j .
The next theorems generalize the results in [22] where the authors assume that In light of the decomposition F = F s + dφ of a vector field into a solenoidal part and a potential part, the conclusion F = dφ of theorem 1.8 can be recast as F s = 0. Therefore theorem 1.8 can be seen as a solenoidal injectivity result in terms of partial data (see [22] and [34,42]). Theorem 1.8 holds also for vector fields F ∈ (S (R n )) n which components are Schwartz functions since in that case (dF) i j ∈ C ∞ (R n ) ∩ A V .

Related Results
There are some earlier unique continuation and partial data results for scalar and vector fields. The partial data problem for scalar fields has a unique solution if f | V vanishes [4,21,25], f | V is polynomial or piecewise polynomial [24,25,47] or f | V is real analytic [24]. A recent partial data result in two dimensions with attenuated X-ray data on an arc can be found in [13]. A complementary result is the Helgason support theorem: if the integrals of f vanish on all lines not intersecting a given compact and convex set, then f has to vanish outside that set [17,43]. The normal operator of the X-ray transform of scalar fields has a unique continuation property under the . This is a special case of a more general unique continuation property of fractional Laplacians [6,15]. There are also partial data and unique continuation results for the d-plane transform of scalar fields when d is odd, including the X-ray transform as a special case d = 1 (see [6] and remark 1.6).
The partial data problem of vector fields is known to be uniquely solvable up to potential fields, if dF| V = 0 [22]. Similarly, the normal operator of the X-ray transform of vector fields has a unique continuation property under the assumptions N 1 F| V = dF| V = 0 [22]. There are other partial data results for vector fields where one knows the integrals of F over lines which are parallel to a finite set of planes [23,39,41] or which intersect a certain type of curve [9,37,45]. There is also a Helgason-type support theorem for vector fields: if the integrals of F vanish on all lines not intersecting a given compact and convex set, then dF vanishes outside that set [22,43].
The normal operator of scalar fields, the normal operator of vector fields and the fractional Laplacian all admit stronger versions of the unique continuation property (see [6,11,12,14,21,22,38,48] and theorems 1.2 and 1.7). Other applications of unique continuation of fractional Laplacians include fractional inverse problems. Especially, the unique continuation of (− ) s is used to prove uniqueness for different versions of the fractional Calderón problem (see e.g. [1,2,[5][6][7]15]).

The X-Ray Transform and Distributions
In this section we define the X-ray transform of scalar and vector fields, and introduce the distribution spaces we use in our main theorems. The basic theory of distributions and Sobolev spaces can be found in [16,18,28,29,44] and the X-ray transform is treated for example in [30,42,43].

Distributions and Sobolev Spaces
The function spaces needed to state our theorems were described in Sect. 1.1.
We let E(R n ) be the space of smooth functions, S (R n ) is the Schwartz space and D(R n ) is the space of compactly supported smooth functions. We equip all these spaces with their standard topologies. The corresponding duals are denoted by E (R n ), S (R n ) and D (R n ). Elements in E (R n ) are identified with distributions of compact support and elements in S (R n ) are called tempered distributions.
We define the space of rapidly decreasing distributions O C (R n ) ⊂ S (R n ) as follows: is the space of polynomially growing smooth functions, i.e. f ∈ O M (R n ) if f and all its derivatives are polynomially bounded. We note that the Fourier transform is an isomorphism F : S (R n ) → S (R n ) and also an isomorphism F : is the set of all continuous functions which decay faster than any polynomial at infinity.
The fractional L 2 -Sobolev space of order r ∈ R is defined as and H r (R n ) becomes a separable Hilbert space for every r ∈ R. It follows that the spaces are nested, i.e. H r (R n ) → H t (R n ) continuously when r ≥ t. One can isomorphically identify H −r (R n ) with the dual (H r (R n )) * for all r ∈ R. We define the following spaces Further, using the Sobolev embedding one can see that f is smooth and f and all its derivatives belong to L 2 (R n ) (see [16,Theorem 6.12]).
The fractional Laplacian is defined as where F −1 is the inverse Fourier transform of tempered distributions. It follows that (− ) s f is well-defined as a tempered distribution for f ∈ O C (R n ) when s ∈ (−n/2, ∞)\Z, and for f ∈ H r (R n ) when s ∈ (−n/4, ∞)\Z (see [6,Section 2.2]). We have that (− ) s : H r (R n ) → H r −2s (R n ) is continuous whenever s ∈ (0, ∞) \ Z and (− ) s also admits a Poincaré-type inequality for s ∈ (0, ∞)\Z (see [6]). We note that (− ) s is a non-local operator in contrast to the ordinary Laplacian (− ). The non-locality implies a unique continuation property (see theorem 1.1 and lemma 4.1) which cannot hold for local operators. We also use local versions of distributions and fractional Sobolev spaces. Let ⊂ R n be an open set and r ∈ R. We denote by D( ), D ( ) etc. the test function and distribution spaces defined in . We define the local Sobolev space H r ( ) as In other words, the space H r ( ) consists of restrictions of distributions f ∈ H r (R n ).
The local Sobolev space is equipped with the quotient norm Then H r ( ) becomes a separable Hilbert space and the restriction map | : H r (R n ) → H r ( ) is continuous. If r ≥ t, then H r ( ) → H t ( ) continuously. One can also isomorphically identify H −r ( ) as the dual ( H r ( )) * for every r ∈ R where H r ( ) is the closure of D( ) in H r (R n ) (see [3] and [28]). If r ≥ 0, then H r ( ) ⊂ W r ( ) where W r ( ) is the Sobolev-Slobodeckij space which is defined by using weak derivatives of L 2 -functions (see [28] for a precise definition). If is a Lipschitz domain, then we have the equality H r ( ) = W r ( ) for all r ≥ 0. More generally, we define the vector-valued test function space (D(R n )) n by saying that ϕ ∈ (D(R n )) n if and only if ϕ = (ϕ 1 , . . . , ϕ n ) and ϕ i ∈ D(R n ) for all i = 1, . . . , n. A sequence converges to zero in (D(R n )) n if and only if all its components converge to zero in D(R n ). We then define the space of vector-valued distributions (D (R n )) n by saying that F ∈ (D (R n )) n if and only if F = (F 1 , . . . , F n ) where F i ∈ D (R n ) for all i = 1, . . . , n. The duality pairing is defined as The test function spaces (E(R n )) n and (S (R n )) n , and the corresponding distribution spaces (E (R n )) n and (S (R n )) n are defined analogously.
The elements in (E (R n )) n are called compactly supported vector-valued distributions. Vector-valued distributions are a special case of currents (continuous linear functionals in the space of differential forms, see [8,Section III]).
For F ∈ (D (R n )) n we define the exterior derivative or curl of F as a matrix which components are (dF) i j = ∂ i F j − ∂ j F i . It follows from the Poincaré lemma (see e.g. [27,Theorem 2.1] and lemma 4.2) that if dF = 0, then F = dφ for some φ ∈ D (R n ) where dφ is the distributional gradient of φ.

The X-Ray Transform of Scalar Fields
Let f ∈ D(R n ) be a scalar field. The X-ray transform X 0 is defined as where γ is an oriented line in R n . When we parameterize the set of all oriented lines with the set the X-ray transform becomes The adjoint or back-projection X * 0 is defined as One then sees that X 0 : D(R n ) → D( ) and X * 0 : E( ) → E(R n ) are continuous maps. Using duality we can define X 0 : E (R n ) → E ( ) and X * 0 : D ( ) → D (R n ) by requiring that where ·, · is the dual pairing. The normal operator is N 0 = X * 0 X 0 and it can be expressed as the convolution Using duality the normal operator extends to a map N 0 : E (R n ) → D (R n ) and the convolution formula holds in the sense of distributions. The normal operator can be seen as the fractional Laplacian (− ) −1/2 up to a constant factor and we have the reconstruction formula where c 0,n is a constant which depends on dimension. Both X 0 and N 0 are also defined for functions f ∈ C ∞ (R n ).

The X-Ray Transform of Vector Fields
Let F ∈ (D(R n )) n be a vector field. The X-ray transform X 1 is defined as where γ is an oriented line. Using the parametrization for oriented lines (see equation (9)) we have We define the adjoint X * 1 as the vector-valued operator One sees that X 1 : (D(R n )) n → D( ) and X * 1 : E( ) → (E(R n )) n are continuous and by duality we can define X 1 : (E (R n )) n → E ( ) and X * 1 : D ( ) → (D (R n )) n by setting We define the normal operator as N 1 = X * 1 X 1 and it can be expressed in terms of convolution The normal operator extends to a map N 1 : (E (R n )) n → (D (R n )) n by duality and the convolution formula holds in the sense of distributions. One has the reconstruction formula for the solenoidal part F s in the solenoidal decomposition F = F s + dφ (see for example [42,43]) where c 1,n is a constant depending on dimension and (− ) 1/2 operates componentwise on N 1 F. Both X 1 and N 1 are also defined for vector fields F ∈ (S (R n )) n .

Partial Differential Operators and Admissible Functions
In this section we introduce constant coefficient partial differential operators and also study the space of admissible functions A V in more detail. A comprehensive treatment of constant coefficient partial differential operators can be found in Hörmander's book [19]. Let us denote by P the set of all polynomials in R n with complex coefficients excluding the zero polynomial P ≡ 0. A polynomial P ∈ P of degree m ∈ N can be identified with the constant coefficient partial differential operator P(D) of order m ∈ N as where D α = D α 1 1 · · · D α n n , D j = −i∂ j and α = (α 1 , . . . , α n ) ∈ N n is a multi-index such that |α| = α 1 + . . . + α n . In fact, using the Fourier transform one sees that where ξ ∈ R n and ξ α = ξ α 1 · · · ξ α n . The polynomial P(ξ ) is also known as the full symbol of P(D). If g ∈ D ( ) where ⊂ R n is an open set, then one can define the distributional derivative P(D)g ∈ D ( ) by duality. Further, it holds that P(D) : H r ( ) → H r −m ( ) is continuous with respect to the quotient norm [29,Theorem 12.15] (see equation (7)).
The set of admissible functions A V which we use in our main theorems can be written as the union where V ⊂ R n is some nonempty open set and H r P, The following proposition implies that the sets H r P,V (R n ) in the union (25) are also Hilbert spaces.

Proposition 3.1
The subset H r P,V (R n ) ⊂ H r (R n ) is a separable Hilbert space for all r ∈ R, P ∈ P and nonempty open set V ⊂ R n .
Proof Clearly H r P,V (R n ) is a linear subspace of H r (R n ). Let f k ∈ H r P,V (R n ) be a sequence such that f k → f in H r (R n ). Then by the continuity of the restriction map | V : is a closed subspace of the separable Hilbert space H r (R n ) and hence itself a separable Hilbert space.

Remark 3.2 We note that in the smooth case we have that E
is a closed subspace of E(R n ) and hence a Fréchet space. More generally, is sequentially closed in D (R n ) under the weak * convergence. These two facts follow from the continuity of P(D) : E(R n ) → E(R n ) and P(D) : D (R n ) → D (R n ) with respect to the standard topologies. More topological properties of kernels of constant coefficient partial differential operators can be found in [46].

Remark 3.3
The interpretation of the condition f ∈ A V is the following. If f ∈ A V , then there is some r ∈ R and some P ∈ P such that f ∈ H r (R n ) and P(D) f | V = 0. The distributional derivatives commute with restrictions, i.e. P(D) f we see that f | V is not only a distribution but in addition f | V ∈ H r (V ) for some r ∈ R. Therefore the existence of r ∈ R and P ∈ P for which P(D) f | V = 0 means that f | V ∈ H r (V ) and f | V is a weak solution to some homogeneous constant coefficient partial differential equation. In other words, for some coefficients a α ∈ C and some integer m ∈ N.
The following proposition is important in the uniqueness of the partial data problem.

Proposition 3.4 The set
Proof Let f 1 , f 2 ∈ A V and λ ∈ C. This means that f 1 ∈ H r 1 (R n ), f 2 ∈ H r 2 (R n ) and P 1 (D) f 1 | V = P 2 (D) f 2 | V = 0 for some r 1 , r 2 ∈ R and P 1 , P 2 ∈ P. It follows that f 1 + λ f 2 ∈ H r (R n ) where r = min{r 1 , r 2 } since the spaces H t (R n ), t ∈ R, are nested vector spaces. We also have that . This implies that

Remark 3.5
The vector space structure of A V is important since it implies that the partial data results we have proved in this article are indeed uniqueness results. Namely, and X 0 ( f 1 − f 2 ) = 0 on all lines intersecting V . Theorem 1.4 then implies that f 1 − f 2 = 0, i.e. the solution to the partial data problem is unique.

Proofs of the Main Theorems
In this section we prove our main theorems. We need a few auxiliary results. The first one is a unique continuation result for fractional Laplacians and the second one is the Poincaré lemma for compactly supported vector-valued distributions.
The proof of lemma 4.2 can be found for example in [20,27]. The third lemma is a known result about the zero set of multivariate polynomials.
Lemma 4.3 is proved in [33] for real coefficients but the result holds also for complex coefficients by splitting b α ∈ C to its real and imaginary parts. We note that the set Z Q is Zariski closed but not the whole space R n . From the coarseness of the Zariski topology (i.e. there are relatively few closed sets) one can already deduce that the set Z Q must be small in topological sense (see e.g. [10,Chapter 15.2]). The next lemma shows how the normal operator of the X-ray transform of vector fields is related to the normal operator of scalar fields (see also [22,Proof of theorem 1.1]).

Proof
The normal operator has the expression Rewrite the kernel as The rest of the results are then direct consequences of theorem 1.1.
Proof of theorem 1.2 By the assumptions we have that f ∈ E (R n ) satisfies P(D) f | V = 0 for some constant coefficient partial differential operator P(D) and ∂ β ((− ) s f )(x 0 ) = 0 for some x 0 ∈ V and all β ∈ N n . Since all the derivatives of (− ) s f vanish at x 0 and the partial derivatives and fractional Laplacian commute, we obtain that Now P(D) f ∈ E (R n ) and we can use corollary 4 on page 652 in [6] to obtain that P(D) f = 0. The claim then follows as in the proof of theorem 1.1.
Proof of theorem 1. 4 The assumption X 0 f = 0 on all lines intersecting V implies that N 0 f | V = 0. Since we also assume that f ∈ E (R n ) ∩ A V or f ∈ C ∞ (R n ) ∩ A V we obtain f = 0 by theorem 1.3.
Proof of theorem 1.7 By lemma 4.4 we have d(N 1 F) = N 0 (dF) componentwise up to a constant factor. Therefore ∂ β (N 0 (dF) i j )(x 0 ) = 0 for some x 0 ∈ V , all β ∈ N n and all i, j = 1, . . . , n. Now by locality of the exterior derivative (dF) i j ∈ E (R n ) ∩ A V . Since N 0 = (− ) −1/2 up to a constant factor we can use theorem 1.2 for the components (dF) i j to obtain that dF = 0. Finally lemma 4.2 implies that F = dφ for some φ ∈ E (R n ).
Proof of theorem 1. 8 The assumption X 1 F = 0 on all lines intersecting V implies that N 1 F| V = 0. Especially d(N 1 F) vanishes to infinite order at some point in V and we can use theorem 1.7 to deduce that F = dφ for some φ ∈ E (R n ).