Differential Geom: Determining Partial Derivatives of f

[k312]

I have been working on determining which partial derivative exists for the surface z=y. i.e. ( partial of f in respect to x, partial of f in respect to y, partial of f in resprct to z). The function f= x^2 -y-z. I think the only ones that exist would be the partial in respect to y and the partial in respect to z since the surface is z=y. Am I on the right track?

 

[Fredrik]

Partial derivatives in differential geometry are always defined using a coordinate system, as in this post. What is your coordinate system?

 

[k312]

x,y,z are the coordinates

 

[k312]

Thanks for the post

 

[Fredrik]

x,y,z are the coordinates

"x,y,z" isn't a coordinate system. Those are just variable names that might possibly represent the components of a coordinate system on a 3-dimensional manifold. Your manifold (the surface defined by the equation z=y) is 2-dimensional.

A coordinate system on an n-dimensional manifold is a function from an open subset of the manifold into $\mathbb R^n$. If this is differential geometry, you need to use a coordinate system. Did you look at the post I linked to to see how the partial derivatives are defined?

 

[Bacle]

I have been working on determining which partial derivative exists for the surface z=y. i.e. ( partial of f in respect to x, partial of f in respect to y, partial of f in resprct to z). The function f= x^2 -y-z. I think the only ones that exist would be the partial in respect to y and the partial in respect to z since the surface is z=y. Am I on the right track?

Just think about it, k312: would the limit quotient make sense in your standard surface.?
What would we mean, by e.g., ||h||->0 , etc. This is why , when we work with manifolds,
most of the work is done in R^n and then brought back/ pulled back into the manifold,
as Fredrik described, by using charts.