3 Let (Y, p) be a metric space. Give an example metric d on Y such that every finction f:(Y, d)→(Y, P) is 3. of horehaby Continuous.

Q: Show that d given by d(x, y) = |x-ylis a metric on R by using definition 4.2.1. You may want to use…

A: we knew  that given metric d(x,y) = |x-y| is always ≥0.  now d(x,y) = 0 ⇔ |x-y| = 0…

Q: Let S be the solid bounded by the planes z = o and y + z = 3, and the cylinder x2+y2 =2y-

A: answer is in next step

Q: LET (X,Y) be a two-dimensional continuous v.a, with density f(x)=x(x+cy), 0<x<1, 0<y<2. So the value…

A: Given:      fx=xx+cy      0&lt;x&lt;1, 0&lt;y&lt;2 To find c, we use…

Q: Ques. 1. Let N be a linear space and d be a metric on N such that d(x + z,y+ z) = d(x,y), x,y € N…

A: I have proved all three conditions gor norm

Q: 3. Prove that if r, y E RP, then ||r + yl|2 = ||x||2 + |ly||² holds if and only if r y = 0. 4. Prove…

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Q: The taxicab metric p on the plane R² is defined by the rule: p((x1, y1), (x2, Y2)) = |x1 – x2| + |y1…

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Q: nclude spaces in between 5. d te In (In y)) at y dy

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Q: Find a transformation that maps the region S = {z = x + iy : x > 0 and –1< y< 1} onto the unit disk…

A: please see the next step for solution

Q: | Use Gauss Divergence theorem to calculate (5zi+ 4yj - 2: k).n dA over the sphere +y+ 9.

A: Here, let F=5x,4y,-2z⇒div F=∂(5x)∂x+∂(4y)∂y+∂(-2z)∂z=5+4-2=7

Q: Verify Divergence theorem for the function F = ēpp over the semi-cylindrical volume V = S x² + y² <…

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

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Q: Verify the Divergonce Theorem for F=(2x-x)+x*y} -aż? taken over the bounded by x -o, X=1, y=0,…

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Q: Let f(x) = 21 - x2 and g(x) = x2 + 3. Use symmetry, if appropriate, to help find the center of…

A: Consider the given two function f(x) and g(x).

Q: A metric space (X,d) such that the function d1(p, q) = (d(p, q)) i not a metric in X

A: Since (X, d) is a metric space. So, d is a function defined as d : X×X→R+∪0 and satisfy the…

Q: Let f(x) = 13 - x? and g(x) = x2 + 5. Use symmetry, if appropriate, to help find the center of…

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Q: Use Gauss Divergence theorem to calculate (5zi+ 4yj- 2z k).n dA over the sphere S: 2 + y? + 22 = 9.

A: This is a problem related to vector calculus. Based on the general formula of gauss divergence…

Q: Let X be a normed space then 1- ||0|| = 0 2-I1-xl = |지l, Vx E X 3- ||x - yl| = |ly - x|,Vx, y E X…

A: As per the policy, we are solving first three subparts. Please resubmit the question and specify…

Q: Show that with the subspace topology Uy on Y, the inclusion map : Y → X is continuous.

A: Given that, UY is subspace topology on Y. Suppose (X, Tx) be topological space and Y is subset of X.…

Q: If a vector-valued function r(t) is continuous at t = a, then the limit r(a+ h) – r(a) lim exist.…

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Q: Verify Gauss Divergence Theorem for F = (x3- yz)i+ (y² - xz)j+ (z3 – xy)k %3D taken over a…

A: GivenF→=x3-yzi^+y2-xzj^+z3-xyk^taken over a rectangular  parallelepiped bounded by 0≤x≤a,0≤y≤b and…

Q: let w- f(xy) and y in Poluy Cordeinate then a+yS %3D or or or 7.

A: Given that, w=fx, y And x and y is in polar form that is, x=rcosθy=rsinθ

Q: 4.) Verify the Divergence Theorem where F(x, y, z)=(x',xz,3z) given by x' + y +z² <4. and E is the…

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Q: (a) If ker(L) has dimension s, what is the rank of L? (b) If ker(L) # {0v}, show that L is not…

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Q: Let V = {(x, y): x 20} with standard operation on R². Is V a vector %3D space?

A: Our objective is to conclude V is a vector space or not.

Q: Evaluate both sides of divergence theorem for A = pa, + pcos()a,+ za, through the closed surface…

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Q: Evaluate. where G is a solid that lies in the first octant bounded by x^2 + y^2 = 3z and z = 3.

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Q: Let X = Rand de f ine ||| on X as ||r|| = Va then X is normed linear space

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Q: Evaluate both sides of divergence theorem for A = pa, + pcos()a , + za, through the closed surface…

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Q: A two dimensional space-time has the metric ds = dv² – vʻdu² . Compute the Ricci tensor Ruv ·

A: Given below clear explanation

Q: Q If X R, define the function ||- ||: R →R*u {o} by ||-|| = |x| + 1. Is (R, ||-||) normed space.…

A: Solution: We will use the following definition: The function .:ℝ→ℝ is a norm if it satisfies the…

Q: Evaluate both sides of divergence theorem for A= pa,+ pcos()a+ za, through the closed surface…

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Q: 4. Show that ||y|| = maxxela.b]ly'(x)| defines a norm for the linear space Y - %3D {y e C' [a, b]: (…

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Q: b) A solid G is bounded by the planes y + z = 4, z = 0, y = 0 and y = 4 – x2. Evaluate dV 4 – Y G

A: Given that  A solid bounded by the planes of y+z=4,z=0,y=0 and y=4-x^2  To evaluate the integral

Q: 4. Verify Divergence Theorem for F =x?i + zj + yzk, taken over the cube bounded by x= 0, x = 1, y =…

A: Gauss divergence Theorem.  Let v be volume bounded a closed surface S and F be a vector…

Q: evaluate.both sides of divergence theorem for A= ra,+rcos(e)a, +ra, through the closed surface…

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Q: Evaluate both sides of divergence theorem for A = pa, + pcos(ø)a, + za, through the closed surface…

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Q: Calculate fff, V.F dV where V is a closed space bounded by x = 0, x = 1, y = 0, y = 1, z = 0 z = 1…

A: In general, triple integrals involve the integral with respect to x, y, and z. Similar to partial…

Q: Ques. 1. Let N be a linear space and d be a metric on N such that d(x + z,y+z) = d(x,y), x,y € N and…

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Q: Evaluate both sides of divergence theorem for A = pa, + pcos($)a+za, through the closed surface…

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Q: Apply Gauss Divergence theorem for F = x²i + y²j + z²k_over the cuboid S bounded by 0 < x< 1, 0< y<…

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Q: Use Green's theorem to compute the circulation of F(x, y) = (-y, x²) around the perimeter of the…

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Q: evaluate both sides of divergence theorem for A= ra,+rcos(8)a , +r²a, through the closed surface…

A: Given problem is :

Q: b. Draw the image of the W-plane into which the region bounded by the straight lines in the Z-plane…

A: The solution is given below

Q: Evaluate both sides of divergence theorem for A = pa,+ pcos(o)a,+ za, through the closed surface…

A: Given A=ρaρ+ρcosϕaϕ+zaz and the closed surface is 0≤ρ≤1, 0≤ϕ≤π, 0≤z≤2. We have to evaluate the both…

Q: Ques. 1. Let N be a linear space and d be a metric on N such that d(x + z,y + z) = d(x, y), x,y E N…

A: Given: N be a linear space and dbe a metric on N such that d(x+z, y+z)=d(x, y),  x, y ∈ N and  d(ax,…

Q: Evaluate / (7xi – zk) · ndA over the sphere S : x² + y² + z² = | S a) by divergence theorem, (b)…

A: a) By divergence theorem, ∬SA. ndA=∫∫∫V∇.A dV Now,  ∇.A=∂7x∂x+∂0∂y+∂-z∂z=7-1=6 By changing to…

Q: Let X=ℝ2 and define d2,:ℝ2×ℝ2→ℝ by d2((x1 ,y1),(x2,y2)) = max{|x1 - x2|,|y1 - y2|}. a) Verify that…

A: Given that X=ℝ2 and defined by d2:ℝ2×ℝ2→ℝ Part(a): Now from given d2:ℝ2×ℝ2→ℝ by…

Q: . Suppose f(a, y) is continuous and non-negative over a non-empty open region R in the plane with a…

A: According to question,f(x, y) is continuous,fx, y ≥ 0        non - negative,∫∫Rfx, y dA = 0

Q: Evaluate both sides of divergence theorem for A= pa, + pcos(0)a , + zaz through the closed surface…

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