What is the difference between $\Delta r$ and $dr$ in Taylor series

Question

All I know about Taylor series is at here. It tells how to expand a funcion to a polynomial. However I see the Taylor series at the form like this (here $r$ is a parametrized surface of $u,v$):

$$\Delta r=dr+\frac12d^2r+o(du^2+dv^2)$$

It confuses me since I don't know the difference between $\Delta r$ and $dr$ (all I know is they are the same at 2 dimensional function, $f=f(x)$ when $\Delta x$ becomes infinitesimal). Can anybody give me a geometrical intuition of $\Delta r$ and $dr$ together with $\Delta^nr $ and $d^nr$ ?

Also, I have heard that in a higher dimension, derivatives exist not implies differentiability. Can anyone give me some examples about that?

Thirdly, I don't know when the 'infinitesimal of higher order' can be neglect. In many case we will neglect the $o(x^n)$ function. But when can we neglect and how can we know which order can we neglect?

Using the above Taylor series as example, let $n$ be an unit normal vector of the points on the surface, my book said

$$\Delta r\cdot n=\left[dr+\frac12d^2r+o(du^2+dv^2)\right]\cdot n=\frac12d^2r\cdot n$$

Why the term $o(du^2+dv^2)$ can be neglect but not the higher or lower order of $o$ funcion?

Answer 1

For a good geometric understanding of the difference between $\Delta r$ and $dr$, look at how a Riemann Summation becomes a Riemann Integral

The width of each column starts as a $\Delta r$ but shrinks to a $dr$ the closer the point spacing.

Answer 2

You might have read that the existence partial derivatives does not imply differentiability. One example would be $$f(x,y) = \dfrac{xy}{x^2+y^2}$$

where $f(0,0) = 0$. It should be clear that $f_x$ and $f_y$ exist and are continuous at $(0,0)$ but the original function $f(x,y)$ is not continuous at zero so it cannot be differentiable there (consider limits as $(x,y)\to(0,0)$.)

Answer 3

Δr represents the secant line connecting two points.

dr represents the tangent line to the curve at the key point.

The difference between the two lines is $$\frac12d^2r+o(du^2+dv^2)$$