Calculating Normal Vector Derivative in Differential Geometry Using Curve Basis

[metzky]

Homework Statement

Using the curve [itex]\vec{a}[/itex](u,v)= (u,v,uv) for all (u,v) ε R^2

Find the matrix for d[itex]\vec{N}[/itex] in the basis of {[itex]\vec{a}[/itex][itex]_{u}[/itex],[itex]\vec{a}[/itex][itex]_{v}[/itex]}

Homework Equations

Well first off i found the partial derivatives
[itex]\vec{a}[/itex][itex]_{u}[/itex] which is 1,0,v, while [itex]\vec{a}[/itex][itex]_{v}[/itex] is 0,1,u
Then using those i found the normal vector which i calculated as [itex]1/\sqrt{v^{2}+u^{2}+1}[/itex] (-v,-u,1)

The Attempt at a Solution

Now this is where i get lost. Our book does not explain this very well at all. It just shows going fron N to dN with no explanation. I tried using the jacobian matrix to calculate the derivative but I'm not sure if this is the right approach. Most of the examples don't have a matrix from so i Know I'm doing something wrong.

The problem is a set from

for reference. I need dN to move on to find the second fundamental forms and so forth

 

[metzky]

Actually could i bring out the 1/sq u^2... out of the jacobian matrix then use the matrix to find the vector or no? so only the -v,-u,1 would be getting partially derived in the matrix?