Non-Vanishing Derivative Functions for Vector Fields X and Y in R^3

In summary, the problem asks to determine whether there exists a function f:R^3-->R with non-vanishing derivative df satisfying Xf=Yf=0, given the vector fields X and Y. In order to solve this problem, one can use the Frobenius integrability conditions, which state that a distribution D={E(x)} on a manifold M is integrable if for all vector fields V,W on M with V(x),W(x) in E(x) for all x in M, [v,w](x) in E(x) for all x in M. This condition can be related to the problem by considering the commutator of the vector fields X and Y, and determining if it satisfies the integr
  • #1
daishin
27
0

Homework Statement


On R^3 with the usual coordinates (x,y,z), consider the pairs of vector fields X,Y given below. For each pair, determine if there is a function f:R^3-->R with non-vanishing derivative df satisfying Xf=Yf=0, and either find such a function or prove that there is none.
(a) X=(e^x)d/dx - ((e^x)z + 2y)d/dz, Y=(e^x)d/dy - (2y)d/dz
(b) X=(e^x)d/dx - ((e^x)z + 2x)d/dz, Y=(e^x)d/dy - (2y)d/dz

Homework Equations



Could you help me start this problem.

The Attempt at a Solution



Sorry I don't know how to start this problem.
 
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  • #2
You'll want to start by looking up Frobenius integrability conditions.
 
  • #3
More hint or idea.

OK I looked at Frobenius integrability condition and still have no clue.
How can I use the integrability condition??
 
  • #4
What is the integrability condition stated in terms of vector fields? You are talking about stuff I haven't looked at for a long time, but isn't the existence of a solution to the PDE's corresponding to these vector fields related to the commutator of the vector fields?
 
  • #5
Frobenius' Theorem

Yes. A distribution D={E(x)} on a manifold M is integrable iff for all vector fiels V,W on M with V(x),W(x) in E(x) for all x in M, [v,w](x) in E(x) for all x in M. But how can I relate this with the problem I asked?
 

1. What is differential geometry?

Differential geometry is a branch of mathematics that studies the properties of curves and surfaces using the concepts of calculus and linear algebra. It involves the study of smooth objects and the ways in which they can be curved or bent.

2. What are the applications of differential geometry?

Differential geometry has many applications in fields such as physics, engineering, computer graphics, and robotics. It is used to model and analyze the behavior of physical systems, design optimal paths for vehicles, and create realistic computer-generated images.

3. What is the difference between differential geometry and regular geometry?

The main difference between differential geometry and regular geometry is that differential geometry deals with smooth and curved objects, while regular geometry deals with rigid and flat objects. Differential geometry also uses the tools of calculus and linear algebra to study these objects, while regular geometry mainly uses geometric constructions and proofs.

4. How is differential geometry related to relativity?

Differential geometry plays a crucial role in Einstein's theory of general relativity. The theory states that gravity is not a force between masses, but rather a curvature of space and time caused by the presence of matter. Differential geometry is used to describe the curvature of space-time and to understand the behavior of objects under the influence of gravity.

5. What are some important concepts in differential geometry?

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