definition of tangent vectors space at point on \(C^\infty\) manifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of tangent vector.
- The reader knows a definition of %field name% vectors space.
Target Context
- The reader will have a definition of tangent vectors space at point on \(C^\infty\) manifold with boundary.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( M\): \(\in \{\text{ the } C^\infty \text{ manifolds with (possibly empty) boundary }\}\)
\( m\): \(\in M\)
\(*T_mM\): \(= \{\text{ the tangent vectors at } m\}\), \(\in \{\text{ the } \mathbb{R} \text{ vectors spaces }\}\) with the operations specified below
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Conditions:
\(\forall r \in \mathbb{R}, \forall v \in T_mM, \forall f \in C^\infty_m (M) ((r v) f = r (v f))\)
\(\land\)
\(\forall v_1, v_2 \in T_mM, \forall f \in C^\infty_m (M) ((v_1 + v_2) f = v_1 f + v_2 f)\)
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2: Note
Let us see that \(T_mM\) is indeed an \(\mathbb{R}\) vectors space.
1st, let us see that the operations are well-defined.
Is \(r v\) really a derivation?
For each \(r' \in \mathbb{R}\), \((r v) (r' f) = r (v (r' f)) = r (r' v (f)) = (r r') v (f) = (r' r) v (f) = r' (r v (f)) = r' ((r v) (f))\), so, \(r v\) is an \(\mathbb{R}\)-linear map.
\((r v) (f f') = r (v (f f')) = r (v (f) f' + f v (f')) = r v (f) f' + r f v (f') = (r v) (f) f' + f (r v) (f')\), so, \(r v\) satisfies the Leibniz rule.
So, \(r v\) is a derivation.
Is \(v_1 + v_2\) really a derivation?
For each \(r' \in \mathbb{R}\), \((v_1 + v_2) (r' f) = v_1 (r' f) + v_2 (r' f) = r' v_1 (f) + r' v_2 (f) = r' (v_1 (f) + v_2 (f)) = r' (v_1 + v_2) (f)\), so, \(v_1 + v_2\) is an \(\mathbb{R}\)-linear map.
\((v_1 + v_2) (f f') = v_1 (f f') + v_2 (f f') = v_1 (f) f' + f v_1 (f') + v_2 (f) f' + f v_2 (f') = (v_1 (f) + v_2 (f)) f' + f (v_1 (f') + v_2 (f')) = (v_1 + v_2) (f) f' + f (v_1 + v_2) (f')\), so, \(v_1 + v_2\) satisfies the Leibniz rule.
So, \(v_1 + v_2\) is a derivation.
1) for any elements, \(v_1, v_2 \in T_mM\), \(v_1 + v_2 \in T_mM\) (closed-ness under addition): which has been seen above.
2) for any elements, \(v_1, v_2 \in T_mM\), \(v_1 + v_2 = v_2 + v_1\) (commutativity of addition): \((v_1 + v_2) (f) = v_1 (f) + v_2 (f) = v_2 (f) + v_1 (f) = (v_2 + v_1) (f)\).
3) for any elements, \(v_1, v_2, v_3 \in T_mM\), \((v_1 + v_2) + v_3 = v_1 + (v_2 + v_3)\) (associativity of additions): \(((v_1 + v_2) + v_3) (f) = (v_1 + v_2) (f) + v_3 (f) = v_1 (f) + v_2 (f) + v_3 (f) = v_1 (f) + (v_2 (f) + v_3 (f)) = v_1 (f) + (v_2 + v_3) (f) = (v_1 + (v_2 + v_3)) (f)\).
4) there is a 0 element, \(0 \in V\), such that for any \(v \in T_mM\), \(v + 0 = v\) (existence of 0 vector): the map that maps each \(f\) to \(0\), \(0\), is obviously a derivation, so, \(0 \in T_mM\), and \((v + 0) (f) = v (f) + 0 (f) = v (f) + 0 = v (f)\).
5) for any element, \(v \in T_mM\), there is an inverse element, \(v' \in T_mM\), such that \(v' + v = 0\) (existence of inverse vector): the map that maps each \(f\) to \(- v (f)\), \(- v\), is obviously a derivation, so, \(- v \in T_mM\), and \((v' + v) (f) = (- v + v) (f) = - v (f) + v (f) = 0 = 0 (f)\).
6) for any element, \(v \in T_mM\), and any scalar, \(r \in \mathbb{R}\), \(r . v \in T_mM\) (closed-ness under scalar multiplication): which has been seen above.
7) for any element, \(v \in T_mM\), and any scalars, \(r_1, r_2 \in \mathbb{R}\), \((r_1 + r_2) . v = r_1 . v + r_2 . v\) (scalar multiplication distributability for scalars addition): \(((r_1 + r_2) . v) (f) = (r_1 + r_2) v (f) = r_1 v (f) + r_2 v (f) = (r_1 v) (f) + (r_2 v) (f) = (r_1 . v + r_2 . v) (f)\).
8) for any elements, \(v_1, v_2 \in T_mM\), and any scalar, \(r \in \mathbb{R}\), \(r . (v_1 + v_2) = r . v_1 + r . v_2\) (scalar multiplication distributability for vectors addition): \((r . (v_1 + v_2)) (f) = r (v_1 + v_2) (f) = r (v_1 (f) + v_2 (f)) = r v_1 (f) + r v_2 (f) = (r v_1) (f) + (r v_2) (f) = (r . v_1 + r . v_2) (f)\).
9) for any element, \(v \in T_mM\), and any scalars, \(r_1, r_2 \in \mathbb{R}\), \((r_1 r_2) . v = r_1 . (r_2 . v)\) (associativity of scalar multiplications): \(((r_1 r_2) . v) (f) = (r_1 r_2) v (f) = r_1 (r_2 v (f)) = r_1 ((r_2 v) (f)) = (r_1 . (r_2 . v)) (f)\).
10) for any element, \(v \in T_mM\), \(1 . v = v\) (identity of 1 multiplication): \((1 . v) (f) = 1 v (f) = v (f)\).
So, \(T_mM\) is an \(\mathbb{R}\) vectors space.
