0708-1300/Term Exam 1: Difference between revisions
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Term Exam 1 |
Term Exam 1 took place on Thursday November 8, 2007, at 6PM, at Sydney Smith 1084. |
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=={{Dror}}'s Internal Notes== |
=={{Dror}}'s Internal Notes== |
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From [http://www.mat.univie.ac.at/~michor/dgbook.pdf Wien] with exercises too. |
From [http://www.mat.univie.ac.at/~michor/dgbook.pdf Wien] with exercises too. |
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==The Exam== |
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===Front=== |
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<center> |
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'''Do not turn this page until instructed.''' |
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'''Math 1300 Geometry and Topology''' |
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'''Term Test''' |
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''University of Toronto, November 8, 2007'' |
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Solve the 4 problems on the other side of this page. |
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Each problem is worth 30 points. |
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You have two hours to write this test. |
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</center> |
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'''Notes.''' |
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* No outside material other than stationary is allowed. |
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* '''Neatness counts! Language counts!''' The ''ideal'' written solution to a problem looks like a page from a textbook; neat and clean and made of complete and grammatical sentences. Definitely phrases like "there exists" or "for every" cannot be skipped. Lectures are mostly made of spoken words, and so the blackboard part of proofs given during lectures often omits or shortens key phrases. The ideal written solution to a problem does not do that. |
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<center>'''Good Luck!'''</center> |
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===Back=== |
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'''Solve the following 4 problems.''' Each problem is worth 30 points. You have two hours. '''Neatness counts! Language counts!''' |
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'''Problem 1 "Compute".''' Let <math>\phi:{\mathbb R}^2_{x,y}\to{\mathbb R}^2_{u,v}</math> be given by <math>u(x,y)=x^2-y^2</math> and <math>v(x,y)=2xy</math>, let <math>f:{\mathbb R}^2_{u,v}\to{\mathbb R}</math> be given by <math>f(u,v)=u^2+v^2</math>, and let <math>\xi\in T_{(0,1)}{\mathbb R}^2_{x,y}</math> be <math>\xi=\partial/\partial x</math>. Compute |
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the following quantities (with at least some justification): |
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# <math>\phi_\star\xi</math>. |
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# <math>\phi^\star f</math>. |
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# <math>(\phi_\star\xi)f</math>. |
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# <math>\xi(\phi^\star f)</math>. |
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'''Problem 2 "Reproduce".''' The tangent space <math>T_0{\mathbb R}^n</math> to <math>{\mathbb R}^n</math> at <math>0</math> can be defined in the following two ways: |
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# <math>T_0^1{\mathbb R}^n</math> is the set of all smooth curves <math>\gamma:{\mathbb R}\to{\mathbb R}^n</math> satisfying <math>\gamma(0)=0</math>, modulo the equivalence relation <math>\sim</math>, where <math>\gamma_1\sim\gamma_2</math> iff <math>\dot\gamma_1(0)=\dot\gamma_2(0)</math>, where in general, <math>\dot\gamma</math> denotes the derivative of <math>\gamma(t)</math> with respect to <math>t</math>. |
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# <math>T_0^2{\mathbb R}^n</math> is the set of all linear functionals <math>D</math> on the vector space of smooth functions on <math>{\mathbb R}^n</math>, which also satisfy Leibnitz' rule, <math>D(fg)=(Df)g(0)+f(0)(Dg)</math>. |
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Prove that these two definitions are equivalent (i.e., that there is a natural bijection between <math>T_0^1{\mathbb R}^n</math> and <math>T_0^2{\mathbb R}^n)</math>. If you use a non-trivial lemma from calculus, state it precisely but you don't need to prove it. |
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'''Problem 3 "Think".''' Let <math>f:M\to M</math> be a smooth function from a compact manifold <math>M</math> to itself. Prove that there is a point <math>y\in M</math> so that <math>f^{-1}(y)</math> is finite. (In fact, there are many such points). |
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'''Problem 4 "Sketch".''' Sketch to the best of your understanding the proof of the Whitney embedding theorem, paying close attention to what is important and little attention to what is not. Here, more than anywhere else, '''neatness and language count!''' |
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<center>Good Luck!'''</center> |
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Revision as of 17:06, 12 November 2007
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Term Exam 1 took place on Thursday November 8, 2007, at 6PM, at Sydney Smith 1084.
Dror's Internal Notes
| Dror's notes above / Student's notes below |
Some Additional Reading
There are some lectures notes of the MIT Open Course Ware. This can be an additional reading for us. There are some exercises too.
More lectures notes from the University of Dublin. This one has exercises.
From Wien with exercises too.
The Exam
Front
Do not turn this page until instructed.
Math 1300 Geometry and Topology
Term Test
University of Toronto, November 8, 2007
Solve the 4 problems on the other side of this page.
Each problem is worth 30 points.
You have two hours to write this test.
Notes.
- No outside material other than stationary is allowed.
- Neatness counts! Language counts! The ideal written solution to a problem looks like a page from a textbook; neat and clean and made of complete and grammatical sentences. Definitely phrases like "there exists" or "for every" cannot be skipped. Lectures are mostly made of spoken words, and so the blackboard part of proofs given during lectures often omits or shortens key phrases. The ideal written solution to a problem does not do that.
Back
Solve the following 4 problems. Each problem is worth 30 points. You have two hours. Neatness counts! Language counts!
Problem 1 "Compute". 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 \phi:{\mathbb R}^2_{x,y}\to{\mathbb R}^2_{u,v}} be 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 u(x,y)=x^2-y^2} 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 v(x,y)=2xy} , 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:{\mathbb R}^2_{u,v}\to{\mathbb R}} be 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 f(u,v)=u^2+v^2} , and 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 \xi\in T_{(0,1)}{\mathbb R}^2_{x,y}} be 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 \xi=\partial/\partial x} . Compute the following quantities (with at least some justification):
- 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 \phi_\star\xi} .
- 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 \phi^\star f} .
- 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 (\phi_\star\xi)f} .
- 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 \xi(\phi^\star f)} .
Problem 2 "Reproduce". The tangent space 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_0{\mathbb R}^n} 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 {\mathbb R}^n} at 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 0} can be defined in the following two ways:
- 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_0^1{\mathbb R}^n} is the set of all smooth curves 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:{\mathbb R}\to{\mathbb R}^n} satisfying 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(0)=0} , modulo the equivalence relation 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 \sim} , where 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_1\sim\gamma_2} iff 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 \dot\gamma_1(0)=\dot\gamma_2(0)} , where in general, 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 \dot\gamma} denotes the derivative 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 \gamma(t)} with respect 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 t} .
- 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_0^2{\mathbb R}^n} is the set of all linear functionals 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} on the vector space of smooth functions 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 {\mathbb R}^n} , which also satisfy Leibnitz' rule, 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(fg)=(Df)g(0)+f(0)(Dg)} .
Prove that these two definitions are equivalent (i.e., that there is a natural bijection between 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_0^1{\mathbb R}^n} 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 T_0^2{\mathbb R}^n)} . If you use a non-trivial lemma from calculus, state it precisely but you don't need to prove it.
Problem 3 "Think". 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\to M} be a smooth function from a compact manifold 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} to itself. Prove that there is 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 M} so 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 f^{-1}(y)} is finite. (In fact, there are many such points).
Problem 4 "Sketch". Sketch to the best of your understanding the proof of the Whitney embedding theorem, paying close attention to what is important and little attention to what is not. Here, more than anywhere else, neatness and language count!
