0708-1300/Class notes for Tuesday, October 9: Difference between revisions
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Class |
==Class Notes== |
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<span style="color: red;">The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.</span> |
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=== Scanned Notes === |
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{{0708-1300/Navigation}} |
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===Typed Notes - First Hour=== |
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'''Reminder:''' |
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1) An immersion locally looks like <math>\mathbb{R}^n\rightarrow\mathbb{R}^m</math> given by <math>x\mapsto(x,0)</math> |
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2) A submersion locally looks like <math>\mathbb{R}^m\rightarrow\mathbb{R}^n</math> given by <math>(x,y)\mapsto x</math> |
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'''Today's Goals''' |
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1) More about "locally things look like their differential" |
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2) The trick Sard's Theorem: "Evil points are rare, good points everywhere" |
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'''Definition 1''' |
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Let <math>f:M^m\rightarrow N^n</math> be smooth. A point <math>p\in M^m</math> is ''critical'' if <math>df_p</math> is not onto <math>\Leftrightarrow</math> rank <math>df_p <n</math>. Otherwise, p is ''regular. |
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'''Definition 2''' |
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A point <math>y\in N^n</math> is a ''critical value'' of f if <math>\exists p\in M^m</math> such that p is critical and <math>f(p) = y</math>. Otherwise, y is a ''regular value'' |
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'''Example 1''' |
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Consider the map <math>f:S^2\rightarrow\mathbb{R}^2</math> given by <math>(x,y,z)\mapsto (x,y)</math>. I.e., the projection map. The regular points are all the points on <math>S^2</math> except the equator. The regular values, however, are all <math>(x,y)</math> such that <math>x^2+y^2 \neq 1</math> |
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'''Example 2''' |
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Consider <math>f:\mathbb{R}^3\rightarrow\mathbb{R}</math> given by <math>p\mapsto ||p||^2</math>. |
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That is, <math>(x,y,z)\mapsto x^2+y^2+z^2</math>. Clearly <math>df|_{p=(x,y,z)} = (2x,2y,2z)</math> and so p is regular <math>\Leftrightarrow df_p\neq 0 \Leftrightarrow p\neq 0</math> |
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So, the critical values are the image of zero, thus only zero. All other <math>x\in\mathbb{R}</math> are regular values. |
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''Note:'' In both the last two examples there were points in the target space that were NOT hit by the function and thus are vacuously regular. In the previous example these are the point x<0. |
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'''Example 3''' |
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Consider a function <math>\gamma</math> from a segment in <math>\mathbb{R}</math> onto a curve in <math>\mathbb{R}^2</math> such that <math>d\gamma</math> is never zero. Thus, rank(<math>d\gamma = 1</math>) and so <math>d\gamma</math> is never onto. Hence, ALL points are critical in the segment. The points on the curve are critical values, as they are images of critical points, and all points in <math>\mathbb{R}^2</math> NOT on the curve are vacuously regular. |
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'''Theorem 1''' |
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''Sard's Theorem'' |
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Almost every <math>y\in N^n</math> is regular <math>\Leftrightarrow</math> the set of critical values of f is of measure zero. |
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''Note:'' The measure is not specified (indeed, for a topological space there is no canonical measure defined). However the statement will be true for any measure. |
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'''Theorem 2''' |
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If <math>f:M^m\rightarrow N^n</math> is smooth and y is a regular value then <math>f^{-1}(y)</math> is an embedded submanifold of <math>M^m</math> of dimension m-n. |
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'''Re: Example 2''' |
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<math>f^{-1}(y)</math> is a sphere and hence (again!) the sphere is a manifold |
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'''Re: Example 3''' |
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<math>f^{-1}(y)</math> for regular y is empty and hence we get the trivial result that the empty set is a manifold |
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'''Proof of Theorem 2''' |
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Let <math>f:M^m\rightarrow N^n</math> is smooth and y is a regular value. Pick a <math>p\in f^{-1}(y)</math>. p is a regular point and thus <math>df_p</math> is onto. Hence, by the submersion property (Reminder 2) we can find a "good charts" thats maps a neighborhood U of p by projection to a neighborhood V about y. Indeed, on U f looks like <math>\mathbb{R}^n\times\mathbb{R}^{m-1}\rightarrow \mathbb{R}^n</math> by <math>(x,z)\mapsto x</math>. |
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So<math> f^{-1}(0) = \{(0,z)\} = \mathbb{R}^{m-n}</math>. ''Q.E.D'' |
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'''Diversion''' |
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Arbitrary objects can be described in two ways: |
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1) With a constructive definition |
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2) with an implicit definition |
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For example, a constructive definition of lines in <math>\mathbb{R}^3</math> is given by <math>\{v_1 + tv_2\}</math> |
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but implicitly they are the solutions to the equations <math>ax+by+cz = d</math> and <math>ez+fy+gz+h</math>. |
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Hence in general, a constructive definition can be given in terms of an image and an implicit definition can be given in terms of a kernal. |
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Homological algebra is concerned with the difference between these philosophical approaches. |
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'''Remark''' |
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For submanifolds of smooth manifolds, there is no difference between the methods of definition. |
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'''Definition 3 ''' |
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Loosely we have the idea that a concave and convex curve which just touch at a tangent point is a "bad" intersection as it is unstable under small perturbation where as the intersection point in an X (thought of as being in <math>\mathbb{R}^2</math>) is a "good" intersection as it IS stable under small perturbations. |
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Precisely, |
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Let <math>N_1^{n_1}, N_2^{n_2} \subset M</math> be smooth submanifolds. Let <math>p\in N_1^{n_1} \bigcap N_2^{n_2}</math> |
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We say <math>N_1</math> is ''transverse'' to <math>N_2</math> in M at p if for <math>T_p N_1\subset T_p M</math> and<math> T_p N_2 \subset T_p M</math> satisfies <math>T_p N_1 + T_p N_2 = T_p M</math> |
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'''Example 4''' |
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Our concave intersecting with convex curve example intersecting tangentially has both of their tangent spaces at the intersection point being the same line and thus does not intersect transversally as the sum of the tangent spaces is not all of <math>\mathbb{R}^2</math>. |
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Our X example does however work. |
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===Typed Notes - Second Hour=== |
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Coming soon to a wikipedia near you. |
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Revision as of 12:12, 9 October 2007
Class Notes
The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.
Scanned Notes
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Typed Notes - First Hour
Reminder:
1) An immersion locally looks like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n\rightarrow\mathbb{R}^m} given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\mapsto(x,0)}
2) A submersion locally looks like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^m\rightarrow\mathbb{R}^n} given by
Today's Goals
1) More about "locally things look like their differential"
2) The trick Sard's Theorem: "Evil points are rare, good points everywhere"
Definition 1
Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:M^m\rightarrow N^n} be smooth. A point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\in M^m} is critical if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df_p} is not onto Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftrightarrow} rank Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df_p <n} . Otherwise, p is regular.
Definition 2
A point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\in N^n}
is a critical value of f if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists p\in M^m}
such that p is critical and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(p) = y}
. Otherwise, y is a regular value
Example 1
Consider the map given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y,z)\mapsto (x,y)} . I.e., the projection map. The regular points are all the points on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^2} except the equator. The regular values, however, are all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y)} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2+y^2 \neq 1}
Example 2
Consider given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\mapsto ||p||^2} . That is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y,z)\mapsto x^2+y^2+z^2} . Clearly Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df|_{p=(x,y,z)} = (2x,2y,2z)} and so p is regular Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftrightarrow df_p\neq 0 \Leftrightarrow p\neq 0}
So, the critical values are the image of zero, thus only zero. All other Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in\mathbb{R}} are regular values.
Note: In both the last two examples there were points in the target space that were NOT hit by the function and thus are vacuously regular. In the previous example these are the point x<0.
Example 3
Consider a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} from a segment in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} onto a curve in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^2} such that is never zero. Thus, rank(Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\gamma = 1} ) and so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\gamma} is never onto. Hence, ALL points are critical in the segment. The points on the curve are critical values, as they are images of critical points, and all points in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^2} NOT on the curve are vacuously regular.
Theorem 1
Sard's Theorem
Almost every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\in N^n} is regular Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftrightarrow} the set of critical values of f is of measure zero.
Note: The measure is not specified (indeed, for a topological space there is no canonical measure defined). However the statement will be true for any measure.
Theorem 2
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:M^m\rightarrow N^n} is smooth and y is a regular value then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(y)} is an embedded submanifold of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M^m} of dimension m-n.
Re: Example 2
is a sphere and hence (again!) the sphere is a manifold
Re: Example 3
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(y)} for regular y is empty and hence we get the trivial result that the empty set is a manifold
Proof of Theorem 2
Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:M^m\rightarrow N^n} is smooth and y is a regular value. Pick a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\in f^{-1}(y)} . p is a regular point and thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df_p} is onto. Hence, by the submersion property (Reminder 2) we can find a "good charts" thats maps a neighborhood U of p by projection to a neighborhood V about y. Indeed, on U f looks like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n\times\mathbb{R}^{m-1}\rightarrow \mathbb{R}^n} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,z)\mapsto x} .
SoFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(0) = \{(0,z)\} = \mathbb{R}^{m-n}} . Q.E.D
Diversion
Arbitrary objects can be described in two ways:
1) With a constructive definition
2) with an implicit definition
For example, a constructive definition of lines in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^3} is given by but implicitly they are the solutions to the equations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax+by+cz = d} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ez+fy+gz+h} .
Hence in general, a constructive definition can be given in terms of an image and an implicit definition can be given in terms of a kernal.
Homological algebra is concerned with the difference between these philosophical approaches.
Remark
For submanifolds of smooth manifolds, there is no difference between the methods of definition.
Definition 3
Loosely we have the idea that a concave and convex curve which just touch at a tangent point is a "bad" intersection as it is unstable under small perturbation where as the intersection point in an X (thought of as being in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^2} ) is a "good" intersection as it IS stable under small perturbations.
Precisely,
Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1^{n_1}, N_2^{n_2} \subset M} be smooth submanifolds. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\in N_1^{n_1} \bigcap N_2^{n_2}}
We say Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1} is transverse to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_2} in M at p if for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_p N_1\subset T_p M} andFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_p N_2 \subset T_p M} satisfies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_p N_1 + T_p N_2 = T_p M}
Example 4
Our concave intersecting with convex curve example intersecting tangentially has both of their tangent spaces at the intersection point being the same line and thus does not intersect transversally as the sum of the tangent spaces is not all of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^2} .
Our X example does however work.
Typed Notes - Second Hour
Coming soon to a wikipedia near you.
