In this final installment on Schleifer’s approach to differential forms and their application to vector calculus, I thought I would take a look at two inter-related items with which Schleifer seems to have had problems. Both of these items deal with using the primitive correspondences to abstractly derive new relations. Before proceeding, it is important to point out that he generally avoids identities involving second derivatives except for the trivial $$\nabla \times (\nabla \phi) = 0$$ and $$\nabla \cdot (\nabla \vec A) = 0$$. As a result, he missed a very interesting opportunity to drive home an important distinction between functions and vectors/forms. As an example of this distinction, reconsider the curl-of-the-curl derived last week.
In vector calculus notation, the curl-of-the-curl identity can be rearranged to give a definition of the Laplacian of a vector
\[\nabla^2 \vec A = \nabla ( \nabla \cdot \vec A) – \nabla \times (\nabla \times \vec A) \; .\]
From the primitive correspondences, the divergence is given by
\[ \nabla \cdot \vec A \Leftrightarrow * d * \phi_A \]
and the curl by
\[ \nabla \times (\nabla \times \vec A) \Leftrightarrow * d \phi_A \; .\]
So the curl-of-the-curl identity can be used to define the differential forms expression for the Laplacian of a vector as given by the correspondence
\[ \nabla^2 \vec A \Leftrightarrow (d * d * – * d * d) \phi_A \; .\]
The natural follow-on question to ask is if this definition is consistent with the expressions for the scalar Laplacian
\[ \nabla^2 f = ( \partial_x^2 + \partial_y^2 + \partial_z^2 ) f \]
introduced in more elementary applications.
Based on the primitive correspondences, the scalar Laplacian naively translates to
\[ \nabla ^2 f \Leftrightarrow * d * d \, f \; \]
So what’s up? Why are they different. Well we can partially fix the difference by subtracting (or adding) a term with the operator $$ d * d * $$ acting on $$f$$ since
\[ d * f = d (f dx \wedge dy \wedge dz ) = 0 \; ,\]
since there is no way for the exterior derivative to raise a 3-form to a higher rank in 3 dimensions, regardless of the functional form of $$f$$.
So we can express the scalar Laplacian as
\[ \nabla^2 f \Leftrightarrow (- d * d * + * d * d) f \; \]
compared to the vector Laplacian
\[ \nabla^2 \vec A \Leftrightarrow (d * d * – * d * d) \phi_A \; .\]
Okay their form is now similar, but why the difference in sign? Well, the sign difference is accepted in the mathematical community and is a consequence of the rank of the form.
The Laplacian of an arbitrary form has the general formula
\[ \nabla ^2 \Leftrightarrow \delta d + d \delta \; ,\]
where the operator
\[ \delta = (-1)^{(np+n+1)} * d * d \; ,\]
where $$n$$ is the dimension of the space, and $$p$$ is the rank of the differential form. If $$n$$ is even then $$\delta = – * d *$$ and if $$n$$ is odd $$\delta = (-1)^p * d * $$. Thus in three dimensions, a $$0$$-form (scalar) has a Laplacian of a different form than a $$1$$-form. We ‘discovered’ this distinction by abstractly manipulating the primitive correspondences (even though Schleifer doesn’t); however, we have found a quantity where the sign differs between the first and the last term. It isn’t clear how this arose or whether it is correct or an artifact of Schleifer’s instance to map vectors and forms so closely. Since he avoids the second derivative identities, we’ll never know what his view on this is.
A related point is the endnote in which Schleifer states that the only type of vector identity he was unable to prove were ones of the form
\[ \nabla \times (\vec A \times \vec B) = (\vec B \cdot) \vec A + \Leftrightarrow *d(*(\phi_A \wedge \phi_B)) \; . \]
Now this statement is a bit of a mystery to me. The cross product has the established mapping
\[ \nabla \times (\vec A \times \vec B) \Leftrightarrow *d(*(\phi_A \wedge \phi_B)) \; ,\]
which expands to
\[ * (\phi_A \wedge \phi_B) = (A_y B_z – A_z B_y) dx \\ + (A_z B_x – A_x B_z) dy \\ + (A_x B_y – A_y B_x) dz \; .\]
For notational convenience define
\[ C_x = A_y B_z – A_z B_y \; , \]
\[ C_y = A_z B_x – A_x B_z \; , \]
and
\[ C_Z = A_x B_y – A_y B_x \; .\]
These three equations can be compactly written as
\[ C_i = \epsilon_{ijk} A_j B_k \]
The curl of this cross-product translates to the language of differential forms as
\[ d * (\phi_A \wedge \phi_B) = C_{x,y} dy \wedge dx + C_{x,z} dz \wedge dx + C_{y,x} dx \wedge dy \\ + C_{y,z} dz \wedge dy + C_{z,x} dx \wedge dz + C_{z,y} dy \wedge dz \; .\]
The pattern exhibited by the coefficients of the two-forms also leads to a compact expression
\[ d * (\phi_A \wedge \phi_B) = \epsilon_{k \ell m} \partial_{\ell} C_k dx^m \; \]
Combining these last two expressions yields precisely what one gets from the classical vector manipulation
\[ \nabla \times (\vec A \times \vec B) = \epsilon_{ijk} \partial_j [\vec A \times \vec B]_k = \epsilon_{ijk} \partial_j \epsilon_{k \ell m} A_\ell B_m \]
which, up to a trivial change of indices, is the same. So why Schleifer says he can’t prove it I don’t know. Perhaps he is trying to say is that in the identity
\[ \nabla \times (\vec A \times \vec B) = (\vec B \cdot \nabla) \vec A – (\vec A \cdot \nabla) \vec B + (\nabla \cdot \vec B) \vec A – (\nabla \cdot \vec A) \vec B \]
there is no proper differential forms mapping of $$(\vec B \cdot \nabla) \vec A$$. Let me say that it even if it doesn’t, it really isn’t a concern. It is not like he actually exploits the abstract language to derive new expressions using the primitive correspondences. If he did he would have discovered the switching sign on the Laplacian.
Overall, I admire Schleifer’s goal but I am concerned about two aspects of this particular program. First is the notational complexity that comes about. For example, $$\vec A \cdot \vec B \Leftrightarrow *(\phi_A \wedge * \phi_B)$$ The need to wrap the expression with parentheses to distinguish $$*\phi_A \wedge * \phi_B \neq *(\phi_A \wedge *\phi_B)$$ makes the notation clunky. Second, and more important, it seems that the desire to continuously re-express the language of differential forms back to vector calculus undercuts the pedagogy and, as shown with the Laplacian, leads to more questions than answers. Hopefully with more time this program can be rescued but for now it doesn’t quite seem ready for prime time.