Curvilinear coordinate, derivative

[kidsasd987]

https://www.particleincell.com/2012/curvilinear-coordinates/
http://www.jfoadi.me.uk/documents/lecture_mathphys2_05.pdf
Hi, I have a question about the curvilineare coordinate system.
I wonder why

is normal to the isosurfaces?isnt ei a tangent vector to the surface ui

since

"With these definitions, we can define the unit vector in the

direction (basis vector)"

 

[Geofleur]

It might help to look at a specific example, and 2D polar coordinates is a good one to start with. Take $u_1 = r$ and draw a picture of $\frac{\partial \mathbf{r}}{\partial r}$ at some point. It points radially away from the origin. Now note that surfaces of constant $r$ are just circles centered at the origin. The gradient is indeed normal to each such surface at the appropriate points.