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FINAL IN PARTIAL DIFFERENTIAL EQUATIONS THALIA D

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FINAL IN PARTIAL DIFFERENTIAL EQUATIONS THALIA D. JEFFRES Instructions: Do your own work. For full credit, your solution must be both correct and complete. If a major computational technique or theorem is used, it should be cited. Sufficient supporting calculations and justifications should be given. You may use all materials that have been created for this course up until the date of the exam. That includes the written notes, your homework solutions and scrap paper, my solutions to homework exercises, any other notes, calculations, and study guides that you have personally created. It also includes the videos I have recorded. You may also use the books that are included in the bibliography for this course, but no others. “Textbook” might be physical or electronic, but in either case means the main (expository) body of the work; it does not include any solution manuals or any other resources that come with the book. If there’s something you don’t understsand, you may ask me. Apart from clarifications that I might make to your or to the class as a whole, no other form of help is allowed. Sign here to indicate that you have complied with this standard of academic integrity: 1. Measure Theory and Integration. Let (?, F, µ) be a measure space, and let f : ? → R be a measurable, real-valued function. Show that the essential supremum of f is less than or equal to the supremum of f. Your solution should demonstrate a knowledge of the definitions of these terms. The definition of the essential supremum of a function appears on page ninety of the notes. Do your work on the remainder of the page. There is also an additional page included in case you need more room. Date: due May 12, 2021. 1 More room for your work on Problem One. 2 2. Fourier Transform. Determine the interaction of the Fourier Transform with the operator TA that we studied in class, under the assumption that A ∈ O(n). In other words, for f ∈ L1 (Rn ), calculate (F(TA f ))(ξ). Recall that TA (f ) = f ? A, and that O(n) is the set of n × n matrices with real entries for which AAT = I. We have done such calculations in the notes: How does the Fourier Transform interact with operators such as derivatives, or the dilation operator, and so on. You can find this material on page 123 and following pages in the notes. We have also done such problems in recent classes. Use the remainder of this page for your work, and there is also another page included if you need more room. 3 More room for your work on Problem Two. 4 3. Multivariable Topics. Consider the change of coordinates - you can think of them as “elliptical polars” - given by x = ar cos θ y = br sin θ for positive constants a and b. (A) Express the first order differential operators ∂/∂r and ∂/∂θ as linear combinations of ∂/∂x and ∂/∂y. (B) Express the Euclidean volume element dA, which in rectangular coordinates would be dA = dx dy, in terms of dr and dθ. Then use this to find the area of the ellipse given by y2 x2 + = C 2. g/l Here, g and l are positive constants. (This ellipse appears in the study of the motion of a pendulum.) We did a calculation like this on pages 194 and 195 of the notes, and we noted at the time that you can use either determinants or the exterior product to find the new volume or area element. Use the remainder of this page for your work on both parts of this problem, breaking up the space as you like. There is also another page if you need it. 5 More room for your work on Problem Three. 6 4. Laplace Operator. Let ? be a bounded domain in Rn with smooth boundary, and take Dirichlet boundary values, meaning that all functions under discussion here satisfy u = 0 on ∂?. Show that if the Laplace operator has real eigenvalues, then these values are negative. You can follow these steps, supplying justifications where necessary. First, suppose ?u = λu, for u not identically equal to zero. Then multiply both sides to get u?u = λu2 . Now integrate over the region ?, and apply the Divergence Theorem (or Green’s Theorem, or Integration by Parts) and finish the proof. You can find the relevant material on pages 204 through 208 of the notes. 7 5. Linear Algebra and Multivariable Topics. Recall that an n × n matrix with real entries belongs to the orthogonal group O(n) if A · AT = I. (A) Give an algebraic proof that if A ∈ O(n), then det A = ±1. (B) Give a geometric proof that if A ∈ O(n), then det A = ±1. (C) Can both values, +1 and −1, actually occur? If you answer in the affirmative, give examples. If you answer in the negative, provide a proof. Do your work for all three parts on the remainder of the page, breaking up the work as you like. 8 J Pages File Edit Insert Format Arrange View Share Window Help 100% GZ Tue 11:47 AM tutor_1-2 < > V 125% v T T + Add Page S SOLUTION: Mat View Zoom Insert Table Chart Text Shape Media Comment Collaborate Format Document Studerpool E Menu Screen Shot 2021-05...4.04 AM A ? ωΕΩ 1. Measure Theory and Integration. Let (12, F,u) be a measure space, and let f:N → R be a measurable, real-valued function. Show that the essential supremum of f is less than or equal to the supremum of f. Your solution should demonstrate a knowledge of the definitions of these terms. The definition of the essential supremum of a function appears on page ninety of the notes. Let f be measurable and there exists CER such that u{f-'(C; +)}=0 By definition we have esssup f(w) - inf fer:u{f-(c;+00)}-0} sup f(w) = a < +00 -=Ø and ae{er:u{8-'(C; +60 )}=0} Note that if we ess sup f(w) s sup f(w) follows that c+0) esssup f(w)= +00 by definition. Since u{f-'(C;+00)}>0 += f''(c;+00)=0 sup f(w)>c sup f(0) = +00 for all ce, we get weh , then f '(a;+5)=0 ? ωΕΩ ωΕΩ Nothing selected. Select an object or text to format. ωΕΩ A What is this A a AN o MAY 5 11 ity 07 ? .. 2