We take a fresh look at the vector differential operators grad, div and curl. There are many vector identities relating these. In particular, there are two combinations that always yield zero results:
Question: Is there a connection between these identities?
Adjoint Operators
A Hilbert space is a function space that is complete and has an inner product: for
and
in
, there is a mapping from
to the real numbers:
. We consider a linear operator
mapping one Hilbert space to another,
.

Linear operator between Hilbert spaces
and
, and its adjoint
.
The adjoint of is defined as the operator
such that
For real-valued functions, we write . In the finite-dimensional case
and
may be represented by a matrix
. Its adjoint is the transformed matrix
.
The differential operator for functions on a bounded interval can be represented by a skew-symmetric matrix
with transform .
In the continuous case, we have no matrix representation and we use integration by parts to obtain the adjoint:
Assuming the boundary term vanishes, this means that
so that , just as in the finite-dimensional case.
Operators on Scalar and Vector Fields
We let be the space of (smooth) scalar fields over
and
the space of smooth vector-valued fields on
. The vector operators
,
and
define mappings between these function spaces, as shown in the diagram:

Vector operators grad, div and curl, mapping between the function spaces and
.
We note that
Clearly, some compositions of these operators are well-defined while others are not. The legal combinations of two operators (right operator first) are
The vector Laplacian, mapping to itself, is defined by
.
For the function spaces and
we define the inner products by integration:
Let us start with the divergence operator, and form :
Assuming that the boundary term (the surface integral) vanishes, this gives
which may be written
This gives us the transpose , and also
(these are reminiscent of
above). We also note that
so the Laplacian operator is self-adjoint.
Now let us look at the curl operator, and form . We use the vector identity
Assuming that the divergence term integrates to zero, this gives
which may be written
This gives us the transpose , showing that the
operator is symmetric.
Finally, using and
, we have
so the two vector identities with which we started,
are actually adjoints of each other.
This answers the question posed at the start of this article:
Sources
Strang, Gilbert, 1986: Introduction to Applied Mathematics. Wellesley-Cambridge Press, 758pp. ISBN: 0-961-40880-4.