# The fundamental misunderstanding of calculus

We all know the fundamental theorems of calculus, if not check Wikipedia.  I now want to  demonstrate what has been called the fundamental misunderstanding of calculus.

Let us consider the two dimensional plane and equip it with coordinates $$(x,y)$$.  Associated with this choice of coordinates are  the partial derivatives

$$\left( \frac{\partial}{\partial x} , \frac{\partial}{\partial y} \right)$$.

You can think about these in terms of the tangent sheaf etc. if so desired, but we will keep things quite simple.

Now let us consider a change of coordinates. We will be quite specific here for illustration purposes

$$x \rightarrow \bar{x} = x +y$$,

$$y \rightarrow \bar{y} = y$$.

Now think about how these effect the partial derivatives. This is really just a simple change of variables.  Let me now state  the fundamental misunderstanding of  calculus in a way suited to our example:

Misunderstanding: Despite coordinate x changing the partial derivative with respect to x remains unchanged. Despite the coordinate y remaining unchanged the partial derivative with respect to y changes.

This may seem at first counter intuitive, but is correct. Let us prove it.

Note hat we can invert the change of coordinate for x very simply

$$x = \bar{x} {-}\bar{y}$$,

using the fact that y does not change. Then one needs to use the chain rule,

$$\frac{\partial}{\partial \bar{x}} = \frac{\partial x}{\partial \bar{x}}\frac{\partial}{\partial x}+ \frac{\partial y}{\partial \bar{x}}\frac{\partial}{\partial y} = \frac{\partial}{\partial x}$$,

$$\frac{\partial}{\partial \bar{y}} = \frac{\partial x}{\partial \bar{y}}\frac{\partial}{\partial x}+ \frac{\partial y}{\partial \bar{y}}\frac{\partial}{\partial y} = \frac{\partial}{\partial y} {-} \frac{\partial}{\partial x}$$.

There we are. Despite our initial gut feeling that that the partial derivative wrt y should remain unchanged we see that it is in fact the partial derivative wrt x that is unchanged.  This can course some confusion the first time you see it,  and hence the nomenclature the fundamental misunderstanding of calculus.

I apologise for forgetting who first named the misunderstanding.