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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

=========================================================================

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

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

=========================================================================

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

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