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# Mathematical Analysis II MA 426

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This 7 page Class Notes was uploaded by Braeden Lind on Thursday October 15, 2015. The Class Notes belongs to MA 426 at North Carolina State University taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/223730/ma-426-north-carolina-state-university in Mathematics (M) at North Carolina State University.

## Reviews for Mathematical Analysis II

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Date Created: 10/15/15

Material to Review for the MA 426 591M Final Steve Schecter December 2 2002 In the following7 when I say you should know a de nition or the statement of something7 that doesn7t mean you should know it word for word it just means that this is something that you should be able to use When I say you should know a proof7 again I don7t mean word for word7 but that you should be able to prove this thing if asked The problems listed are ones you should be able to do Whenever the book mentions metric spaces in de nitions7 theorems7 etc7 you can substitute 1 Topology of IR 0 O O O O 16 and 17 Euclidean space7 norms7 inner products7 metrics Consider these sections as background 21 Open sets De nition 211 proof of Proposition 212 proofs that 90 E IR H 90 11gt 17 ac E R2 x1 gt 07 and ac E R2 901 lt 1 are open proof of Proposition 213 problems 211 and 213 22 Interior of a set Omit 23 Closed sets De nition 2317 statement of Proposition 232 Proofs from lecture that various sets are closed 24 Accumulation points De nition 2417 statement of Theorem 242 second problem assigned Aug 26 see httpcoursesncsueduma426lec003assignhtml 25 Closure De nition 2517 statement of Proposition 2527 problem 252 26 Boundary De nition 2617 proof of Proposition 2627 Example 263 27 Sequences De nition 2717 statement of Propositions 272 and 2737 proof of Proposition 2747 statement of Proposition 2767 example 2777 problem 273 as we did it 28 Completeness You should know what a Cauchy sequence is and the state ment of Theorem 285 o 29 In nite series Omit 2 Compactness7 path connectedness7 continuity 0 31733 Compactness The following material is from lectures we didn7t follow the text a De nition of a bounded set b Statement of Nested Sets Property as done in lecture c Statement of Bolzano Weierstrass Theorem as done in lecture d Open cover de nition of compactness7 problems 1737 57 6 assigned Sept 13 see httpcoursesncsueduma426lec003assignhtml o 41 Continuity De nition 4117 the rephrasing of De nition 4127 and De nition 413 especially the second sentence statement from lecture of Easy Theorem 41477 and proofs that 8 ii7 iii 8 i7 and 8 iii for functions from R to R proof that fx1ac2 x1 is continuous7 and use of this result to do problem 3 on p 108 problem 413 o 42 Images of compact sets under continuous maps Proof of Theorem 421 both the convergent subsequence proof and the open cover proof item 43 Operations on continuous mappings Proof of Theorem 431 for f R a R and g R a RP statements of Proposition 432 and Corollary 433 o 44 Boundedness of continuous functions on compact sets Statement of Theorem 441 347 427 45 Path connected sets De nition7 proof that D07 6 is path connected7 proof of Theorem 421 for path connected not connected sets Problem 4217 452 0 o 46 Uniform continuity De nition 4617 statement of Theorem 462 0 51 Pointwise and uniform convergence De nitions 511 and 512 the lemma proved in lecture for showing that a sequence fk converges uniformly proof of Proposition 514 problems 5117 5127 513 3 The derivative 0 61 De nition of the derivative De nition 611 Problems 612 and 614 p 384 problem 2 proof that if fac Ax b then DF A 62 Matrix representation De nition 621 Statement of Theorem 622 Prob lems 6217 622 o 63 Continuity of differentiable functions Proofs of the following 1 if L R a R is linear7 then there is a constant M such that Lac Hg M ac 2 if f A a R is differentiable at 900 then f is continuous at 900 O o 64 Conditions for differentiability The example that begins the section gure 641 Statement of Theorem 641 De nition 642 Example 643 Problem 642 as we did it o 65 Differentiation rules Proof of the multiplication by a constant rule77 and the sum rule77 Statement of the chain rule Theorem 651 Relation of the chain rule to the directional derivative formula Example 654 as done in lecture Problems 6527 653 o 67 Mean value theorem Proof of Theorem 671 rst part Problem 675 o 68 Second derivative and Taylor7s Theorem The idea of the derivative of Df Statements of Theorems 682 and 6837 and of the version of Theorem 685 that we did in lecture Problem 685 o 69 Maxima and minima Statement of Theorem 692 Proofs of Theorem 6947 and of the lemma we used to prove it Problems 6937 696 4 lnverse and implicit function theorems o 71 lnverse Function Theorem Statement of Theorem 711 Problems 7117 7127 7157 and p 442 problem 25 o 72 lmplicit Function Theorem Statement of Theorem 721 Pp 4397440 problems 47 67 8 o 77 Lagrange multipliers Omit 5 Integration 0 81 Basic de nitions7 statement of Riemann7s Criterion 813 o 82 Volume Statement of equivalent condition for volume 0 given in lecture Proofthat the graph of a continuous function f I a R has volume 0 Problems 2 and 6 assigned Nov 19 see httpcoursesncsueduma426lec003assignhtml o 83 Statement of our simpli ed version of Lebesgue7s Theorem 0 84 Statement of properties to v of the integral Problems 37 47 5 assigned Nov 19 o 93 Change of variables Omit Material to Review for the Second MA 426 Test Steve Soheoter November 7 2002 In the following7 when I say you should know a de nition or the statement of something7 that doesn7t mean you should know it word for word it just means that this is something that you should be able to use When I say you should know a proof7 again I don7t mean word for word7 but that you should be able to prove this thing if asked The problems listed are ones you should be able to do Whenever the book mentions metric spaces in de nitions7 theorems7 etc7 you can substitute 0 O O O 51 Pointwise and uniform convergence De nitions 511 and 512 the lemma proved in lecture for showing that a sequence fk converges uniformly proof of Proposition 514 problems 5117 5127 513 61 De nition of the derivative De nition 611 Problems 612 and 614 p 384 problem 2 problem 641 as we did it proof that if fx Ax b then Dfx A 62 Matrix representation De nition 621 Proof of Theorem 622 Problems 6217 622 63 Continuity of differentiable functions Proofs of the following 1 if L IR a R is linear7 then there is a constant M such that Lx Hg M ac 2 if f A a R is differentiable at 900 then f is continuous at 0 64 Conditions for differentiability The example that begins the section gure 641 Statement of Theorem 641 De nition 642 Example 643 Problem 642 as we did it 65 Differentiation rules Proof of the multiplication by a constant rule77 and the sum rule7 Proofs of properties of O and 0 done in lecture and homework Proof of the chain rule Relation of the chain rule to the directional derivative formula Example 654 as done in lecture Problems 6527 6537 655 O O 67 Mean value theorem Proof of Theorem 671 rst part Problem 675 68 Second derivative and Taylor7s Theorem The idea of the derivative of Df Statements of Theorems 682 and 683 The version of Theorem 685 that we did in lecture7 and its proof Problem 685 69 Maxima and minima Statement of Theorem 692 Proofs of Theorem 6947 and of the lemma we used to prove it Problems 6937 696 71 lnverse Function Theorem Statements of Contraction Mapping Theorem and Theorem 711 Problems 7117 7127 7157 and p442 problem 25 72 lmplicit Function Theorem Statement of Theorem 721 Pp 4397440 problems 4 6 8 7 7 77 Lagrange Multipliers Omit MATERIAL TO REVIEW FOR THE FIRST MA 426 TEST STEVE SCHECTER In the following when I say you should know a de nition or the statement of something that doesnt mean you should know it word for word it just means that this is something that you should be able to use When I say you should know a proof again I dont mean word for word but that you should be able to prove this thing if asked The problems listed are ones you should be able to do Whenever the book mentions metric spaces in de nitions theorems etc you can substi tute R O O O O O 16 and 17 Euclidean space norms inner products metrics Consider these sections as background 21 Open sets De nition 211 proof of Proposition 212 proofs that 90 E R H 90 11gt 1 ac E R2 x1 gt 0 and 90 E R2 901 lt 1 are open proof of Proposition 213 problems 1 and 3 22 Interior of a set Omit 23 Closed sets De nition 231 statement of Proposition 232 Proofs from lecture that various sets are closed 24 Accumulation points De nition 241 statement of Theorem 242 second problem assigned Aug 26 see httpcoursesncsueduma426lec003assignhtml 25 Closure De nition 251 statement of Proposition 252 problem 5 26 Boundary Omit 27 Sequences De nition 271 statement of Propositions 272 and 273 proof of Proposition 274 statement of Proposition 276 example 277 problem 3 as we did it 28 Completeness You should know what a Cauchy sequence is and the statement of Theorem 285 29 In nite series De nition 291 proofs that if Eyck ac and Eyk y then 2xk yk 90 y and 209019 0x statement of Theorem 292 proof of Theo rem 293 statement of Comparison Test problems 3 and 4 assigned Sept 6 see httpcoursesncsueduma426lec003assignhtml 31733 Compactness The following material is from lectures we didn7t follow the text 1 De nition of a bounded set 2 Statement of Nested Sets Property as done in lecture 3 Statement of Bolzano Weierstrass Theorem as done in lecture 4 Open cover de nition of compactness problems 173 5 6 assigned Sept 13 see httpcoursesncsueduma426lec003assignhtml 41 Continuity De nition 411 the rephrasing of De nition 412 and De nition 413 especially the second sentence statement from lecture of Easy Theorem 41477 and proofs that 8 ii iii 8 i and 8 iii for functions from R to 1 AAA O O O STEVE SCHECTER Rm proof that fx17x2 x1 is continuous7 and use of this result to do problem 3 on p 108 problem 3 42 Images of compact sets under continuous maps Proof of Theorem 421 both the convergent subsequence proof and the open cover proof 44 Boundedness of continuous functions on compact sets Statement of Theorem 441 Proof that if K is compact and 90 is a point7 then there is a point in K that is closest to 90 43 Operations on continuous mappings Proof of Theorem 431 for f R a R and g R a RP statements of Proposition 432 and Corollary 433 347 427 45 Path connected sets De nition7 proof that D07 6 is path connectedl7 proof of Theorem 421 for path connected not connected sets Sec 427 problem 1 46 Uniform continuity Omit

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