Parametrized Set ... McInerney, Example 3.3.3

Peter

Well-known member
MHB Site Helper
I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.3: Geometric Sets and Subspaces of $$\displaystyle T_p ( \mathbb{R}^n )$$ ... ...

In Section 3.3 McInerney defines what is meant by a parametrized set ... and then goes on to give some examples ...

... see the scanned text below for McInerney's definitions and notation ...

I need help with Example 3.3.3 which reads as follows:

In the above example we read the following ... ...

"... ... The parametrized set $$\displaystyle S = \phi (U)$$ is the plane through the origin described by the equation $$\displaystyle 2x - 3y - z = 0$$ ... ... "

Can someone please demonstrate how/why the parametrized set $$\displaystyle S = \phi (U)$$ is the plane through the origin described by the equation $$\displaystyle 2x - 3y - z = 0$$ ... ... ?

Help will be much appreciated ... ...

Peter

*** EDIT ***

Reflecting on the above question we have $$\displaystyle \phi (u, v) = ( u, v, 2u - 3v )$$ ...

... so ... taking variable $$\displaystyle (x, y , z)$$ in $$\displaystyle \mathbb{R}^3$$ ... ..

... then for $$\displaystyle u = x, v = y$$ we have $$\displaystyle z = 2x - 3y$$ .

But how exactly (rigorously) is $$\displaystyle z = 2x - 3y$$ the same as $$\displaystyle \phi (U)$$ ... ?

I am not happy with the above rough thinking/reasoning ...

Peter

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So that readers will understand McInerney's approach to parametrized sets and the relevant notation ... I am providing the relevant text at the start of Section 3.3 as follows ... ...

Hope that helps,

Peter

Last edited:

Evgeny.Makarov

Well-known member
MHB Math Scholar
Well, the set $\{(x,y,2x-3y)\mid x,y\in\mathbb{R}\}$ is the plane $2x-3y-z=0$. More precisely, it contains coordinates of points in this plane.