2. Let A{1,2,3} be equipped with the discrete metric d, (i.e. d(x, y) = 1 if x # y, and0 otherwise), and let the Cartesian product E = [0,1] × A be equipped with the%3Dd(x, y)do-metric (i.e. d»((x, a), (x', a')) = max{|r – x'|, d(a, a')}). Give examples of%3D%3Da) a compact subspace K CE,b) a non-compact subspace LC E,c) a connected subspace M CE,d) a disconnected subspace N C E, and justify your claims.

Discrete metric: Let X be a nonempty set. The discrete metric is defined as follows, dx,y=1 if x=y0…

2. Let A = d(x, y) do-metric (i.e. d»((x, a), (x', a')) = max{|x – x'|, d(a, a')}). Give examples of {1,2,3} be equipped with the discrete metric d, (i.e. d(x, y) = 1 if r y, and = 0 otherwise), and let the Cartesian product E [0, 1] x A be equipped with the a) a compact subspace K C E, b) a non-compact subspace LCE, c) a connected subspace M CE, d) a disconnected subspace NCE, and justify your claims.

2. Let A = d(x, y) do-metric (i.e. d»((x, a), (x', a')) = max{|x – x'|, d(a, a')}). Give examples of {1,2,3} be equipped with the discrete metric d, (i.e. d(x, y) = 1 if r y, and = 0 otherwise), and let the Cartesian product E [0, 1] x A be equipped with the a) a compact subspace K C E, b) a non-compact subspace LCE, c) a connected subspace M CE, d) a disconnected subspace NCE, and justify your claims.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: the function is a metric of R: d(x,y) = sqrt(x^2−y^2); x,y ∈ R

A: let z1=(x1,y1) and z2=(x2,y2) now, d(X x Y, d) is a metric space if 1)d(z1,z2)=0 if z1=z2 now, if…

Q: 6. f(x, y) = (x - 1)² + (x −y)¹. − Ans.: global minimum 0 at (1,1).

A: Given function is f(x, y) = (x-1)4 + (x-y)4

Q: Suppose f(): R" → R (n EN). (a) What is directional derivative of f(-) at x ER" in direction v ER"?…

Q: 5 - If f(x) = За and the equation of the normal line to the curve at the point (a, b) is x + 2a y =…

A: The objective is to the find the value of equation a+b-c=?. We are given that the equation of…

Q: The maximum rate of change of the function f(x, y, z) = xyz at the point (1, 2, 3) occurs in the…

A: In this question we have to find the correct answer.

Q: a) Sketch the graph of 7(t) =and show the direction of the increasing t, where 0 c) If 7(1)…

A: The vector field is rt=cost, t, sint. Now, if yt=t, then x=cosy and z=siny. Thus, the vector field…

Q: 15) Calculate the the curl of divergence and

A: To find Divergence and curl of a given vectors

Q: Find the tangent plane (Xo, Yo, Zo) from f (x, y) = x² - 3y²; (0,1, -3)

Q: P(E) = 0.55, P(F) = 0.35, P(E U F)= 0.8 Find: a)P(E ∩ F) b) P(E^c) c) P(E^c…

Q: За and the equation of the tangent line to the curve at the point (a,b) is y = 2 - If f(x) = -x- x +…

Q: If the circle x² + y² + 2gx+2fy + c = 0 bisects the circumference of the circ x² + y² + 2g′x + 2ƒ'y…

A: It is given that the circle x2+y2+2gx+2fy+c=0 bisects the circumference of the circle…

Q: 6) What is the tangent plane of f (x, y) = 5x² – 2xy? + 3xy + 4x at point (xo, Yo) = (1, –2) ?

A: The question is solved in step. 2

Q: Let u= u(x,t) be the solution. Which of the following is one of the characteristics of the pde A)…

Q: $d$ is a metric on $X$. Show that $\rho : X^2 \to [0,\infty)$ defined by \[ \rho(x,y) =…

Q: Evaluate -I doc ezx ď'cax +2) where d'cax) is the derivative of S lx) delta function

A: Given : ∫-∞-1dx e2xδ'(x+2)

Q: Suppose z = z(x,y) is x + y+z = e¬(x+y+z) , then which relation is correct? az D): az az az az 1 ду…

A: The solution is given as follows :

Q: af of H.W: Find and for the functions: b əx ду 1. f(x, y) = (xy-1)² 3. f(x, y) = (x²-1)(y + 2)…

A: Given functions are 1. f(x, y)=(xy-1)2 3. f(x, y)=(x2-1)(y+2) 4. f(x, y)=x(x2+y2) To Find: ∂f∂x…

Q: Let be a curve in the xy-plane. Find d'y dx² -|(x, y)=(1, 1) = _ 6x³3 + 19y³ = 25 તે પુ da²(x,…

Q: For a function y = f(x), show that the curvature is given by к(x) = Hint: Parameterize the function…

Q: Evaluate d:/dx on the surface x: + + x²y®=² = 3 at the point (1, 1, 1).

Q: Find an expression for a unit vector normal, no, to the surface x = (4 – cos (v)) cos (u), y = (4 –…

A: The complete solutions are given below

Q: =

Q: 4. 다음 벡터함수가 연속이 되는 범위를 구하여라. (1) r(t) = sint'i+ cos tj+ ik t- 2 (2) r(t) = e"'i+ In (t +3)j+ k %3D…

Q: Assume that R (t) + 0. Show that T'(1) = R' (t) – (De||R (t)|)Ť(t) T'(t : ||T (t)||

A: As per the question (and the general conventions in vector calculus) we have a vector valued curve…

Q: If z = (xy)/(x-y), show that x^(2)(∂^(2)z)/(∂x^(2))+2xy(∂^(2)z)/(∂x∂y)+y^(2)(∂^(2)z)/(∂y^(2))=0.

A: Given- z = xyx - y To Show- Show that x2∂2z∂x2 + 2xy∂2z∂x ∂y + y2∂2z∂y2 = 0

Q: Find the expression of the metric ds? = (dx')? + (dx²)² + (dx³)² and it's 1st and 2nd components of…

Q: Use the definition of the directional derivative (using limits!) to compute f~u(x,y) for f(x,y) =…

A: Given Data: Let us consider the given data: v=-3,4. To find: fux,y

Q: Evaluate ∮C (x + 3y)dx + ydy where C is the Jordan curve given by the graphs of y = e^x, y = e^−x…

A: ∮Cx+3ydx+ydy Where C:y=ex, y=e-x, y=e-1

Q: Be z = (x-y)/(x+y), where x = (u/v), y = (v2/u), calculate ∂z/∂u and ∂z/∂v

A: The given equation is z=x-yx+y where, x=uv, y=v2u. Find the value of ∂z∂u and ∂z∂v. Here we use the…

Q: d (x, y) = (x - y)^2 does the function define a metric o

Q: 1 (vz,xz,xy) dr where C is the union of the curve y=x* from (0,0,0) to 1+xyz Evaluate (11,0) and the…

Q: Given the vectors x and y',..., y™ in R". Our objective is to check whether x E conv{y', . , y™}.…

A: Given:vectors x and y1,...,ym in Rn

Q: Consider the following problem from Science: A mitochondria with velocity vector r' (t) starts out…

Concept explainers

Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0). The multivariable limit of this function as it approaches the points (x0, y0) and is denoted as:

Question

2. Let A
{1,2,3} be equipped with the discrete metric d, (i.e. d(x, y) = 1 if x # y, and
0 otherwise), and let the Cartesian product E = [0,1] × A be equipped with the
%3D
d(x, y)
do-metric (i.e. d»((x, a), (x', a')) = max{|r – x'|, d(a, a')}). Give examples of
%3D
%3D
a) a compact subspace K CE,
b) a non-compact subspace LC E,
c) a connected subspace M CE,
d) a disconnected subspace N C E, and justify your claims.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Step 7

Step by step

Solved in 7 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Limits and Continuity$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

I really need help figuring out how to do this problem, using the cheat sheet formulas needed for the equation
7. Determine aT, aN and k for the space curve = (e' cos t – e' sin t)i + ((v5)e')j+ (e' cos t + e' sin t)k.
Let C be the closed curve indicated in the diagram. Use Green’s Theorem to calculate (look at the following picture)
For a function y = f(x), show that the curvature is given by к(x) = Hint: Parameterize the function as r(x) = (x, f(x)). |ƒ"(x)| (1 + (f'(x))²) ³/2*
BE 4SIGAT LLDEH' A 21: dz ax 22: Find the derivative of f(y, x) = xe?y + cos(3yx) at a point (1,0,7)in the direction of V = -i+ 3j + k 23. -* B.JPG coy) ata po 33p0 III
1. Compute the curvature of the curve y=x^2+4x+1 at the point on the curve when x=1. Give the exact answer.
Suppose f(): R" → R. Define (a) The directional derivative f'(x; v) at x in direction v. (b) When does f(-) have a gradient at x? (c) When is f(-) proper? (d) When is f() lower semicontinuous? (e) When is f(-) convex? Just answers B, D, and E please :)
If the circle x² + y² + 2gx + 2fy + c = 0 bisects the circumference of the circle x² + y² + 2g'x + 2fy+ c' = 0, , then prove that 2g'(g-g')+2ff- f')=c-c'