In his 1887 paper , Weingarten announced an elegant formula, found in Equation (58) below, for the jump discontinuities in the second derivatives of the Newtonian potential across the boundary of a self-gravitating body. Recall that the Newtonian potential induced by the mass density distribution over a domain is given by
where is the radius vector and the symbols and denote the functions and of the coordinates .
Our goal is to construct formulas analogous to Equation (58) for the jump discontinuities in the potentials induced by simple and double layer distributions over a surface S. A simple layer potential is usually associated with a charge distribution while a double layer potential is usually associated with a dipole distribution . They are given by the surface integrals
where denotes the normal derivative. The relevant elements of potential theory can be found in the classic texts  - .
2. The Relevant Elements of Tensor Calculus
Let us briefly summarize our notation, as well as the basic elements of tensor calculus that we will use in this paper. Refer the three-dimensional Euclidean space to general curvilinear coordinates . The Latin indices will run the values 1, 2, and 3. Treat the radius vector as a function of the coordinates denoted by . We will typically collapse the arguments of functions and denote simply by . Let
be the covariant basis and
be the covariant metric tensor. The contravariant metric tensor is defined as the matrix inverse of . We will use the standard elements of tensor calculus in the Euclidean space including the multiplication and contraction of tensors, index juggling, and covariant differentiation. For the necessary background in tensor calculus, we especially recommend  as well as the book  by Tracy Thomas who was one of the key figures in the theory of compatibility conditions for discontinuous tensorial fields. We also recommend our books   from which we borrow the notation and some of the elementary results with regard to the compatibility conditions.
Now consider a surface S. Let the Greek indices run the values 1 and 2 and refer the our surface S to the coordinate . Suppose that
are the equations of the surface S, where the symbol represents the functions . Let be the surface restriction of the ambient radius vector function . Then is given by the composition
The shift tensor is given by
The surface covariant basis, given by
is related to the ambient covariant basis by the identity
The surface covariant metric tensor is given by
Its matrix inverse is the contravariant metric tensor . All objects on the surface are subject to the surface covariant derivative denoted by .
The shape of the boundary S is characterized by the unit normal and the curvature tensor , also known as the second fundamental form. These important tensors satisfy the equations
where the last equation is known as Weingarten’s equation. Finally, we will make frequent use of the projection formula
although it will almost always appear with different combinations of index names.
3. One-Sided Limits and Jump Discontinuities
Consider a field defined in the ambient space surrounding the interface S referred to surface coordinates . Suppose that S divides the space into two regions denoted by and . Consider a point P on S with ambient coordinates and surface coordinates . Then let be the limiting value of as from the region and, similarly, be the limiting value of as from the region . The fields and defined on the surface S are known as one-sided limits of and are illustrated in Figure 1.
The same concepts can be applied to tensor fields such as , where the symbols and are understood as the one-sided limits of .
Figure 1. Ilustration of the concept of a one-side limit.
(Note that the ambient derivative cannot be applied to and since the latter are defined only on the surface.)
Recall that the surface covariant derivative satisfies the chain rule
when applied to fields defined in the surrounding ambient space. Therefore,
assuming that the covariant derivative and the limit can be interchanged. This is known as the Hadamard lemma. A rigorous derivation of this lemma can be found in .
For a field F that is discontinuous across S, the quantity is called the jump discontinuity and, according to notation introduced above, is denoted by , i.e.
From Equations (19) and (20), we find that
Contract both sides of this equation with , i.e.
Since (see  and  )
Exchanging the names of the indices i and j and solving for , we find
This formula “decomposes” the jump discontinuity of the derivatives in terms of the jump discontinuity of F itself and the jump discontinuity of the normal components of . The latter can be thought of as the jump discontinuity in the normal derivative of F.
Let denote the jump discontinuity in the n-th normal derivative of F. In other words,
and so on. With the help of these symbols, Equation (26) can be rewritten as
Equation (30) is called the first-order compatibility conditions. The Nth-order compatibility conditions are an expression for the jump discontinuities in the Nth-order derivatives of F in terms of the . In essence, compatibility conditions treat the quantities as the degrees of freedom governing the jump discontinuities and thus provide a great deal of structure and insight.
The goal of the next section is to derive the general second-order compatibility conditions which can then be applied to the simple and double layer potentials and .
4. The General Second-Order Compatibility Conditions
Our present goal is to derive the compatibility conditions for the discontinuity jumps in the second derivative of F. With the help of the projection formula
can be rewritten as
By the chain rule,
We will now analyze each of the terms on the right separately.
4.1. The Term
Another application of the projection formula
4.2. The Term
Let us once again use the project formula
Continuing with the product rule, we have
Since and , we find
The combination is often denoted by , i.e.
With the help of this new symbol, is given by
4.3. The Final Relationship
Combine the two terms obtained in the preceding sections:
Collecting like terms we arrive at the final second-order compatibility conditions
Let us call attention to several important special cases. If the field F is continuous across S, i.e. , then
If F has continuous first-order derivatives across S, i.e. , then
Finally, if F is continuous along with its derivatives, i.e. , then the compatibility conditions have the particularly simple form
Let us also use the main compatibility condition (43) to derive the general expression for the jump discontinuity in the Laplacian . Raising the index j and contracting with i, we find
Recall that , , and . Therefore,
Meanwhile, for the invariant , we have
where the invariant is known as the mean curvature. Thus, finally, we have
This completes the main technical challenge of our investigation, that is, to derive the general compatibility conditions. In order to apply the obtained formulas to specific situations, we must simply determine the values of the surface fields , , and . For the classical Newtonian potential, as well as the potentials associated with simple and double layers, these fields can be determined by analyzing the well-established partial differential equations and boundary conditions that govern them. This task will be accomplished in the next three sections.
5. The Classical Newtonian Potential and Weingarten’s Condition
Recall that the Newtonian potential V is defined by the integral equation
where is the domain occupied by the mass distribution , as illustrated in Figure 2.
The potential V satisfies the Poisson equation
inside and the Laplace equation
Figure 2. A self-gravitating body with density and boundary S.
inside . Furthermore, V is continuous across S along with its derivatives, i.e.
In other words,
Thus, from equation
we observe that
As a result, Equations (51), (52) tell us that
Having determined , , and , we can conclude from Equation (46), that
which is precisely Weingarten’s condition announced in .
6. Simple Layers
Recall that potential induced by the simple layer distribution on the surface S is defined by Equation (2). Similarly to the Newtonian potential V, is twice differentiable away from S where it satisfies the Laplace equation
On the other hand, the continuity conditions across the interface S are distinct from those of the Newtonian potential V. Specifically, while is continuous across S, i.e.
its first derivative experiences a jump discontinuity captured by the equation
Next, from equation
it follows that
Since from Equation (59), it is apparent that
we conclude that
Once again, having determined , , and , we can use the compatibility conditions (44) to arrive at the equation
This is our main result concerning the simple layer potential .
In conclusion, we note that for the special case of a flat interface S, characterized by and therefore and , we have
7. Double Layers
Recall that the double layer potential is defined by the integral Equation (3). Similar to , the potential is also twice differentiable away from the interface S where Laplace equation
away from the boundary S. However, at the boundary S, is discontinuous. Its jump discontinuity is described by the equation
Its normal derivatives, on the other hand, are continuous, i.e.
In other words
In order to determine , let us once again turn our attention to equation
from which we observe that
Since, according to Equation (69), the Laplacian is zero on either side of the boundary, and therefore we find that
Thus, from Equation (43), we arrive at the compatibility condition
This is our main result concerning the double layer potential .
As before, note the special case of a flat interface, i.e.
The three most popular distributions in the classical potential theory are the volumetric distribution, described by the singular three-dimensional Integral (1), the simple layer, described by the two-dimensional singular surface Integral (2), and the double layer, described by the two-dimensional singular Integral (3). The second derivatives of the resulting potential experience finite jump discontinuities at the corresponding interfaces. For the volumetric distribution, the second-derivative discontinuity is given by Equation (58) established by Weingarten  in 1887. In this paper, we provided the analogous relationships (67), (76) the simple layer and double layers.