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.