Analysis II MATH 522
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This 2 page Class Notes was uploaded by Zechariah Hilpert on Thursday September 17, 2015. The Class Notes belongs to MATH 522 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 29 views. For similar materials see /class/205293/math-522-university-of-wisconsin-madison in Mathematics (M) at University of Wisconsin - Madison.
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Date Created: 09/17/15
TOPICS COVERED IN THE SECOND MIDTERM MATH 522 SPRING 2009 1 Differentiation inside an integral when can we differentiate wrt t the integral of a function q5z t if we integrate wrt 1 Notice that I did a simpli cation of the Theorem in the book so it is better to look at the notes As usual you will need to check hypothesis if you apply the Theoremi 2 De nition of integral a De nition of the integral of a continuous function over kcells as iteration of integrals in one variable b Theorem showing that the integral of continuous functions over kcells does not depend on the order of integration using StoneWeierstrass c De nition of integral over other compact domains by extending it trivially to a cell containing the domain Theorem showing that for these discontinuous functions the integral is still well de ned using the glue function 3 Change of variable formulai a De nition of primitive functions and ips b Theorem stating that a C1 map between open set of R with F0 0 and FO invertible can be decomposed into composition of primitive functions and ips the fancy row reduction77 process i c De nition of partition of unity associated to an open cover of a compact set d Change of variable Theoremi Learn the steps in the proof how do we apply all the previous concepts in this proof Be ready to prove this theorem in special cases for example when is a primitive function 4 Differential forms and surfaces their de nitions algebra of kforms how do you add multiply by functions and elementary properties i 5 Basic kforms and the Theorem showing that if a form is written in terms of basic forms it is zero iff each coef cient is zero 6 The wedge product and how to use it The differential and how to apply it to kformsi Properties of the differential d2 0 and the product rule including the proof of why d2 0 which I could ask 7 Stokes theoremi Effect of a pullback in forms in their wedge product and in their differentiation Effect of the composition of two pullbacks in the forms and their wedge producti Learn well how pullbacks worki Effect of pullbacks in the integral of kformsi Af ne simplexes and their orientation Effect of a change of orientation on the integral over the simplex The oriented boundary of an af ne simplexi Af ne chains and boundaries of af ne AA 0quot 93 VV AA CL 0 VV a1 i Differentiable Chains and their boundaries Be sure to know examplesi f Statement of Stokes Theorem and how to use it A 39D V PRACTICE PROBLEMS This list is not exhaustive it is meant only for practice and to give an idea of the level of the questions 1 Consider the function x A Jew Md 0 u 3 Calculate 051 If you are going to apply any Theorem be sure to check its hypothesis 2 Suppose that a function 45 ab A lR is continuous Prove that 211 Ab dz Ab 1 3 Let uvu gt 01 gt 0 and de ne 39i39 U 7 R2 by uw u2 7 v2uvi These are called hyperbolic coordinates Let S l9 X 2 Calculate 12 yQWEdy 5 Do not worry about the precise number you get i 4 If w f is a zero form an A gdz Where dz is a basic kforrn prove that dwA deAerAdAl 5 If 39i39 is a differentiable chain With zero boundary in an open set V C RT and if 9 is an n 7 1 form on V show that d9 rnust vanish at some point x 6 Vi 6 Let 39i39 be the differentiable chain given by 39i39 12 X 0 2 7gt R2 TJ 7 cos 9 Tsint9i Find the oriented boundary of 39i39 and be sure to include a drawing Let w Edy ydzi Calculate wl 31 Same question for 3 zdy 7 ydzi 7 Let 39i39 be a surface de ned by 39i39 01 gtlt 01 7 IRS ts gth1sgth2sgth3s Where g h1h2h3 are all C20 Let 51dydzydzAdzzdIAdyi 5 1 Without using Stokes Theoremi b Is there a solid V in R3 With nonzero volume such that 8V 39i39 Where 8V is the positively oriented boundary of V Explain 8 Consider the differentiable chain a torus a 027r X 2 7r 7 R3 a Calculate au v a 12 cos u cos 1 a 12 cos u sin 1 bsin Calculate Ba and prove that if w is any C2 lforrn then w0i
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