Recall our main theorem about vector fields. Theorem. Let R be an open region in E2 and let F be a C1 vector field on R. The following statements about F are equivalent: (1) There is a differentiable function f : R → R such that ∇f = F. (2) If C is a piecewise C1 path in R, then C F · dx depends only on the endpoints of C. (3) C F · dx = 0 for every piecewise C1 simple, closed curve in R. Furthermore, statements (1)–(3) imply (4) curl F = 0, and (4) implies (1)–(3) when R is simply connected
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Math 245C Homework 1 Yunbai Cao 904066974 Apr 11, 2014 Exercise 1.10.3 Proof. Since R equipped with the usual topology F is Hausdorff, (R, F ) is Haudorff as F is stronger than F. Given any x ∈ R, for any y ∈ R\{x}, there exists By open such that By ∩ Byx = ∅. Then R\{x} = is open. Let K = [0, 1]\Q and L = { 1 }. Then K c = (−∞, 0) ∪ (1, ∞) ∪ Q ∈ F , so K 2 is closed. And L is closed by previous paragraph. Let U ⊃ K, V ⊃ L be two open neighbourhoods of K, L. Claim: U ∩ V = ∅. As F is generated
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Sequence - 26 (Use First) Precision - 29 (Use First) Technical - 29 (Use First) Confluent - 20 (Use as Needed) Sequence: Sequence is my "use first" for learning, and I use this Learning Pattern first. I utilize this when it comes to my job, I use a very methodicaly system. I always get with the intel shop, prior to planning any missions. After the intel dump, I then go look to see what other teams have been to this venue. If they have, then I look at their routes and times they have used. Because
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size geographically. Is it a network that is in a room or small office building? Or do the systems attached to the network span the entire length of a state? Local Area Network or LAN is two or more electronic devices that are connected with in a limited area such as a room or building. The evolution of LANs has changed as technology has become more powerful. Early local area networks used coaxial cable as a means to communicate. Then in the 80s twisted pair starting with Cat3 cable made the
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1. Preliminaries 1. Set operations A set is a collection of some items such as outcomes of an experiment. We denote sets using upper case letters, say A and write a ∈ A if a is an element belonging to A. If A and B are two sets, then the notation A ⊆ B means that the set A is included in B , i.e. each element of A is also an element of B : A⊆B iff [∀a ∈ A : a ∈ B ] If A ⊆ B and A = B then we say that the set A forms a proper subset of B and write A ⊂ B . Dorota M. Dabrowska (UCLA)
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Math 5616H Midterm 1 with solutions Spring 2013 March 8, 2013 Total 80 points 1. (15 points) Let f (x) and g(x) be real continuous functions on an interval [a, b], such that b b f 2 (x) dx = a a b g 2 (x) dx = 1. Prove that a f (x)g(x) dx ≥ −1, and that a b f (x)g(x) dx = −1 if and only if f ≡ −g on [a, b]. Answer: Since f and g are continuous, so is (f + g)2 , which is therefore integrable. We compute: b b b b b 0≤ a b [f (x)+g(x)]2 dx = a f (x)2 dx+2 a
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November 20, 2011 Academic and Professional Success UNIV103-1104B-276 My Foundations Labs lets you evaluate your academic foundational skill set so you can self-motivate to enhance your strengths and strengthen your weak areas. We completed sections in Math, Reading and Writing in My Foundations Lab. I completed assignments in Math, Reading and Writing sections of My Foundations Lab. I started off by first doing the Reading section of the assignment first. I chose this section first
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_ _ _ _ _ _ _ _ _ _ i _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ b _ _ _ _ _ _ _ _ ,_ _ _ _ _ _ _ _ _ _ _-_ _ i _ _ _ _ _ _ _ _ _ b _ _ _ _ _ _ _ _ _ _ _ _ _ _. Questions: Answers: 1. y=/2x-1/ – what is its Range A. R(-∞,5/2)U(5/2,∞) in set notation? J. D(5,∞) 2. y=5x+15 – what is its Domain C. R(-∞,1/2)U(1/2,∞)
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User Manual Critique The user manual is for the Samsung GT-S5250 mobile device. It is a user manual with instructions, operation, and maintenance of the Samsung Wave smart phone. In order to approve or disapprove of this user manual, the manual will be critique in six criteria: Organization, order of information, access features, variety features, graphics, and content. Provide positive and negative examples from the online manual that represent each aspect of your designated criteria.
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Qualtrics Offices: A Big Eye Opener People take cues from external stimuli about their expected behavior. From the clothes that are worn, to the car one drives, to the words one uses around you – all of this tends to influence one’s attitude and behavior. One’s physical workspace is another big determinant of HOW one works. Reflecting upon the experience visiting Qualtrics, I believe the special physical environment they’ve created positively influences their level of collaboration, teamwork
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