4. Show that the closed interval [0, 1] and the unit circle S¹ = {(x, y) | x² + y² = 1} cannot be homeomorphic by considering connectivity. (Hint.For any continuous f : [0, 1] → S¹, what can you say about the image of (0, 1)?)

Suppose there exist a homeomorphism f:[0, ∞)----->Rthen restriction of f f|[0,∞)\{0} : [0,…

Answered: 4. Show that the closed interval [0, 1]…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2
3. Let A = {1,2,3), let S = P(A) and let RC S x S be a relation defined by the rule (X,Y) E R (XnY=0) ^ (XUY = A). (a) Prove or find a counterexample for the following properties: • antisymmetry. asymmetry. symmetry. irreflexivity/ • reflexivity. • transitivity • uniqueness
2. Given the function f(x,y)=x²+2y² +5sin(2x+3y): Set both x and y intervals as [-4, 4] and 100 gridlines • Format axes: x [-5, 5], y [-5, 5], 2 [-5, 18], 2 clip -5 to 18 Set the Focus Point to (0, 0, 9). Make colors and transparency to your liking Remember to hit "enter" after changing the settings
b). Show that J(-ydx+xdy) = x₁Y₂ −X₂Y₁ where is the line segment joining (x₁,₁) and (x₂,₂). [Verify using Mathematica
1. Consider the set S: S = {(x, y, z, w) = R¹ : x² + y² + z² +w² = 1,x+y=1} (a) Show that any point in S can be written locally as the graph of a C¹ function. (b) Which point on the surface S is the farthest away from the point (0, 0, 2, 0)?
You have been given the graphs below, which one of them represents f(x,y)=9-x²-y² ? 1) 2) 3) 4) 30 16 14 12 10 20 -15 5) 6) 7) 04 -20
) Let f(z) = (z − ₁)(z − 3₂)... (zn), where ₁, 2,...,n E C. If y is a positively oriented simple close contour closing every ; in its interior, prove that $ dz ƒ(z) = 0.
Exercise 4.1. Let F : R² → R² be the map defined by F (G) Describe the image by F of the points lying on the unit circle centered at 0, i.e. { (4) e R² | x² + y² = 1}. 3y) for any (G) E R².
5. Show that the following orthogonal families of curves: are x² + y? : = ax, x² + y? = by
Let C = C₁ U C2 U C3, where C₁ is the line segment from (-1, −1) to (0, 0), C2 is the line segment from (0, 0) to (−1, 1), and C3 is the portion of the circle x² + y² = 2 from (−1, 1) to (-1,-1) traced counterclockwise. 1 x Define F(x, y) =. 2 √x+2 1+y² a. Show that I is conservative by finding all its potential functions. b. Based on (a), what is the value of SF · dR?
5. Let C be a circle, and let P be a point not on the circle. Prove that the maximum and minimum distances from P to a point X on C occur when the line XP goes through the center of C. [Hint: Choose coordinate systems so that C is defined by x² + y? = r2 and P is a point (a, 0) on the x-axis with a ±r; use calculus to find the maximum and minimum for the square of the distance. Don't forget to pay attention to endpoints and places where a derivative might not exist.]
Answer bothh

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4. Show that the closed interval [0, 1] and the unit circle S¹ = {(x, y) | x² + y² 1} cannot be homeomorphic by considering connectivity. (Hint. For any continuous f : [0, 1] → S¹, what can you say about the image of (0, 1)?)