logistic_guy
Senior Member
- Joined
- Apr 17, 2024
- Messages
- 1,538
here is the question
Z(x,y)=(x,xy,y).
Show the mapping Z:R2→R3 is a patch.
Definition
the definition don't say anything about regularity
to solve this i check first if Z is injective then to check the regularity condition that is the Jacobian matrix have full rank. if i solve the Jacobian matrix how to know it have full rank?
Z(x,y)=(x,xy,y).
Show the mapping Z:R2→R3 is a patch.
Definition

Patch -- from Wolfram MathWorld
A patch (also called a local surface) is a differentiable mapping x:U->R^n, where U is an open subset of R^2. More generally, if A is any subset of R^2, then a map x:A->R^n is a patch provided that x can be extended to a differentiable map from U into R^n, where U is an open set containing A...
mathworld.wolfram.com
the definition don't say anything about regularity
to solve this i check first if Z is injective then to check the regularity condition that is the Jacobian matrix have full rank. if i solve the Jacobian matrix how to know it have full rank?