v1i2


Volume 1, Issue 2

Letters and Notes

Exploring Alternate Notations for
Partial Differentials
Jonathan Bartlett
DOI: 10.33014/issn.2640-5652.1.2.bartlett.2

Recent work has shown that differentials can be made
more algebraically manipulable if the notation is improved
(Bartlett and Khurshudyan, 2019). Can this also be the
case with the Jacobian notation for partial differentials (i.e.,
∂y
∂x )?

The typical counterexample to why partial differentials
could not be algebraically manipulable comes from an ex-
ample similar to the following. Take a three-variable equa-
tion, such as y = x2z. This equation has six partial deriva-
tives. For this example, we will note that ∂y

∂x = 2xz and
∂y
∂z = x2. If partial differentials were algebraically manip-
ulable, we should be able to find ∂z

∂x by simple algebraic
manipulation:

∂z
∂x
=

∂y
∂x
∂y
∂z

=
2xz
x2 =

2z
x

However, actually calculating the partial derivative of z
with respect to x directly yields − 2z

x .

As was the case in Bartlett and Khurshudyan (2019), the
reason for the contradiction is not a failure of the power
of differentials, but merely in the notation. To understand
this, let us look at where partial differentials come from.
Taking our original example, let us simply take the total
differential.

y = x2z

d(y) = d(x2z)

dy = x2 dz + 2xz dx.

Let’s say we wanted the partial derivative ∂y
∂x . A partial

derivative between two variables means that no variation is
allowed to happen in the other variables. In this case, the
other variable is z. If no variation happens in z, that means
that dz must be zero. Therefore, we set dz = 0 and solve

for the ratio of the other differentials, like this:

∂y = x2 (0) + 2xz ∂x
∂y = 2xz ∂x
∂y

∂x
= 2xz

Now, when we solve for ∂y
∂z , we do the same process, but

set a different differential to zero (i.e., dx). Note that both
of these derivatives involve ∂y, but they come from two
different modifications to the original equation. Therefore,
the ∂y in ∂y

∂z is not the same ∂y that is in ∂y
∂x . Indeed, the

bottom differential gives us additional information about
which ∂y is being discussed. That is why the notation ∂y

∂x
cannot be separated—information about the numerator is
contained in the denominator, and is therefore lost when
the fraction is split.

To resolve this situation, we merely need a notation that
allows us to be more specific about which partial differen-
tials we are dealing with. Here we will describe two pos-
sible approaches, each with different implications. I don’t
think either of these are the “best” system, and I hope that
this discussion sparks additional ideas which combines the
advantages of each system. It is also possible that the no-
tation chosen to represent partial differentials is based on
your own goals of how to manipulate them.

The two systems will be based on subscripting our partial
differentials. In system 1 (for lack of a better name), we
will subscript the partial differentials with the variable that
was allowed to freely move. In the system 2, we will sub-
script the partial differentials with the variables that were
forced to not move.

So, in system 1, instead of ∂y
∂x , we will now modify the no-

tation to include the variables which were allowed to freely
move. Therefore, the partial derivative of y with respect
to x will be ∂x y

∂x x
. Likewise, the partial derivative of y with

respect to z will be ∂z y
∂z z

. This notation can be further sim-
plified by noting that, if a variable is allowed to freely vary,
that is the same as being a total differential. Therefore,
∂x x = dx. This simplifies our derivatives to ∂x y

dx and ∂z y
dz .

This also means that a total derivative of a dependent vari-
able is simply the sum of its partial differentials. dy =
∂x y + ∂z y In fact, we can actually carry pieces of the par-



78 Letters and Notes

tial derivative around in the subscripts, by saying ∂x+z y =
∂x y + ∂z y. This can have benefit in large, multivariable
equations, as the subscript can simply be added to as ad-
ditional partials are added in. When all of the variables
are included, then the partial differential is the same as the
total differential.

The drawback to system 1 is that it only fully works when
y is a dependent variable, and all other variables are inde-
pendent variables. If the other variables are have dependen-
cies among them, the system breaks down. For instance, if
x = f (z), then ∂x x is not equal to dx, and dy is not equal to
the sum of its partials, depending on the particular formula
those partials come from.

The other system, system 2, takes this into account by,
instead of subscripting which variables are allowed to freely
vary, subscripts which differentials were forcibly set to zero.
This system makes fewer requirements of the equation itself
(because it specifies what we are doing to the equation),
but also lends itself, as far as I can currently tell, to fewer
simplification mechanisms.

Therefore, the partial derivative of y with respect to x
(where dz = 0) would be specified as ∂z y

∂z x
. This can get

unweildy in equations with a large number of variables, as
you would need to subscript every variable whose differen-
tial which was set to zero.

In any case, both system 1 and system 2 allow for extend-
ing algebraic manipulability to partial differentials. The
advantage of system 1 is that it has rules for interchang-
ing partial and total differentials, but the disadvantage that
you must know a priori which variables are dependent and
independent. The advantage of system 2 is that you do
not need the knowledge of which variables are dependent
and independent, but building up total differentials from
partial ones is more complicated.

Unfortunately, the notation for system 1 and system 2
are identical, so it would be confusing to use them both.
Another alternative would be to use set notation. So, using
system 1, you could say ∂d∈x y

∂d∈x x
, while system 2 would say,

∂d!z y
∂d!z x

. Basically, this uses set notation to say which ele-
ments were allowed to be modified. There are drawbacks
here, too, as it makes it look like system 1 and system 2
are combinable, but I am not sure that they are. Addition-
ally, the notation itself is quite unweildy.

The primary point of this exercise is to show that (a) the
problem of the algebraic manipulability of partial differen-
tials is primarily a notational problem, and (b) to start the
conversation about what notation might possibly replace
the current notation, and the benefits and drawbacks of

each option.

Bartlett, J L and A Zh Khurshudyan (2019). “Extending
the Algebraic Manipulability of Differentials”. In: Dy-
namics of Continuous, Discrete and Impulsive Systems,
Series A: Mathematical Analysis 26.3, pp. 217–230.

The Unlearnable Checkerboard
Pattern
Eric Holloway
DOI: 10.33014/issn.2640-5652.1.2.holloway.1

In the history of machine learning, a famous upset was the
disproving of the perceptron’s generality by use of the XOR
problem. There is no linear separator than can classify
the points with 100% accuracy (Russell and Norvig, 2009,
pg. 741).

Figure 1: Difficulty of Linear Separation in an XOR

x1

x2

In response, the multilayer perceptron was invented, which
eventually became the modern day neural network.

However, the neural network, and other machine learning
paradigms, still have a problem learning a tiled XOR pat-
tern called the checkerboard, see Figure 3.

The checkerboard classifications can be generated using a
logic expression. To generate the classifications from 2N
variables, we use the logic expression x1 ⊕ xN+1.

Here is an example with 4 variables, x1, x2, x3 and x4. The
logic expression is x1 ⊕ x3.