Differential Geom: Determining Partial Derivatives of f

In summary, it seems that the only partial derivatives that exist for the surface z=y are the partial derivatives in respect to y and z.
  • #1
k312
3
0
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?
 
Physics news on Phys.org
  • #2
Partial derivatives in differential geometry are always defined using a coordinate system, as in this post. What is your coordinate system?
 
  • #3
x,y,z are the coordinates
 
  • #4
Thanks for the post
 
  • #5
k312 said:
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 [itex]\mathbb R^n[/itex]. 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?
 
  • #6
k312 said:
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.
 

1. What is differential geometry?

Differential geometry is a branch of mathematics that deals with the study of curves, surfaces, and other geometric objects using techniques from calculus and linear algebra. It also involves the study of how these objects change and interact with each other under various transformations.

2. What is a partial derivative?

A partial derivative is a mathematical tool used to calculate the instantaneous rate of change of a function with respect to one of its variables, while holding all other variables constant. It is denoted by ∂f/∂x, where f is the function and x is the variable of interest.

3. How do you determine the partial derivatives of a function?

To determine the partial derivatives of a function, you first need to identify the variables in the function. Then, you take the derivative of the function with respect to one variable, treating all other variables as constants. This process is repeated for each variable, resulting in multiple partial derivatives.

4. What is the chain rule in differential geometry?

The chain rule in differential geometry is a rule used to find the derivative of a composite function. It states that the derivative of a composite function is equal to the product of the derivatives of its component functions.

5. How are partial derivatives used in differential geometry?

Partial derivatives are used in differential geometry to study the behavior of geometric objects under transformations. They allow us to calculate the rates of change of these objects along different directions, which is crucial in understanding their properties and relationships with other objects.

Similar threads

  • Differential Geometry
Replies
9
Views
332
  • Differential Geometry
Replies
3
Views
3K
  • Differential Geometry
Replies
2
Views
511
  • Differential Geometry
Replies
12
Views
3K
  • Differential Equations
Replies
1
Views
1K
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
301
Replies
1
Views
58
Replies
4
Views
594
  • Calculus and Beyond Homework Help
Replies
3
Views
702
Back
Top