3. i) Let (X, d) be a metric space. If æn → x in (X, d) and y e X, prove that lim d(xn, y) = d (x, y) . %3D 00

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 X (2,3. 4) and de fined metrie on X as d(x, y) = |-y Find open ball B (2.2) = (0) O (2) 0 2.0…

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Q: 3. Evaluate [[[ 20 av G if G is a solid in the first octant bounded by the plane y + z = 2 and the…

A: Solution :

Q: Exercise 11. iz Let f(z) and CR be the upper half circle with center (0,0) and radius R z5+1 taken…

A: Given problem is :

Q: Given the region under the plane 2x + 3y + z = 6 that lies in the first octant. 0 24. What is the…

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Q: Let A = { (x,y)e RXR; 0<max{}x-31. ly-21)<1}, then: %3D None of the choices OA is not compact…

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Q: Consider the following mappings on RXR: * d1(1, y) = e²=y, d2 = |2r - 2y). and d3 = V12r – yl. %3D…

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Q: Consider the following mappings on RXR: d1(x,y) = e*-Y, d2(x,y) = |2x –yl and d3(x,y) V=yl O Both d1…

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Q: Q2: (a) Let M = {x,y,z) and define d: MXM → R by Determine whether d is a metric on M. d(x,x) = d(y,…

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Q: If u = x°f(x/y) according to Euler's theorem xUx + yuy is 5u %3D

A: given that u=x5fxy verify whether xux+yuy=5u is true or false

Q: Suppose that Di is the open disk with center 0 and radius 1. Let f(z) be analytic on Di \ {0} and…

A: Given- Suppose that D1 is the open disk with center 0 and radius 1. Let fz be analytic on D1\0 and…

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: Let f function from (R,d) to (R,d ) defined as f(x) = x , where d is .the discrete metric then f is…

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Q: Let I = , zdV ,where D is bounded by the planes x = 0, z = 0, x+ y = 2 and the cylinder x2 + z2 = 1…

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Q: Let d: Rx R*→[0, ∞) such that d(XF)=√(x, y)² show that (R.d) metric space.

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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: View S' as the boundary of the 2-disk D² = {(x, y) E R² | x² + y? < 1}. Fix To E S', and let X = (S'…

A: Given : D2=x,y∈ℝ2|x2+y2≤1 where x0 belongs to S1 which the boundary of 2-disk. X=(S1×S1)∪(D2×x0) To…

Q: and y have density flay)=82y tor Find ECX14) and ECYIX) リフズ7679 %3D

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Q: Exercise. a) For every ,r, y E R, let p(r, y) =| x² – y? |. Is pa metric U..... on R? b) For every…

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Q: Let X=R2 and defined d2:  R2 x R2 to R by d2((x1, y1)) = max{|x1-x2|, |y1-y2|} Verify that d2 is a…

A: Given a function d2: R2×R2 → R, defined by d2x1,y1,x2,y2=maxx1-x2,y1-y2. To verify that d2 is a…

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: Suppose that D is the unit disk in the xy-plane, which is contained in the unit square S = [0, 1] ×…

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Q: Given the vecor-valued function R(t) = cosht i ln(t+1) j+et k, find lim h→0 R(0+ h) - R(0) h O…

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Q: Suppose d is Show that p: X?> [0,00) defined by a metric on X, p Cxy)= 3d (x,y) %3D 2+3d(x, u) is…

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Q: (b) Let X = {r € R: -1 <1<0or 0 <1< 1} with the usual metric.Then X is disconnected.

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Q: 3.4 Give the definition of a separable met rie space (S, d). 3.5 Show that the metrie space (R, d),…

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Q: At what points (x,y.z) in space are the tunctions continuous? a. h(x.y.z) = In (4z² - 4x – 2y³ - 5)…

A: This is a problem of multi variable calculus.

Q: (10) Define the metric d on R² by d( (x₁, y₁), (x2, y2)) = max{ |x₁ − x2\,|y₁ - y2|}. Verify that…

A: The given problem is to verify that the given metric d on R2. Given metric…

Q: or tho gonal Use the least, me thad to find the squones projetiom of bz Ġ 2 -a anto the salwmin of…

A: Consider the given:         A=011211and  b=12-2T=12-2

Q: Let X = R and x, y E R. Admit for all x and y the metric d(x, y) = |arctan x arctan y|. | Compare…

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Q: O If and exist in a neighborhod of (a, b) and are continuous at (a, b) then f(x, y) is…

A: This is a problem of multivariable calculus.

Q: Let f(x,y,z) = zIn(x³ycos(z)), find Z 3x² O In(x3ycos(z)) - ztan(z) 0 2 Y о 3z X af ila

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Q: -) Let -y M(x, y) = x2 and N(x, y) = + y? x2 + y? how that ƏN ƏM dx ду ) Show that -y F(x, y) = x² +…

A: Given :-  A vector field for conservative field in the plane ,  ∂N∂x = ∂M∂y To Determine :-  M x,y =…

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: 2. Assume that (X, dX) and (Y, dY ) are complete spaces, and give X × Y themetric d defined byd((x1,…

A: To prove that every Cauchy sequence in the product space XxY converges (under the metric defined in…

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

A: Based on given data,For Non empty open region R,fx, y is continuous,andfx, y is non -…

Q: If X =R and d(x,y)= 1, then X not metric semi metric metric space

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Q: Let W be the region that is cut aut from x't v<4 by the pora blae id Rs X'ty ond the plane =10.…

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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: Let R be the region in the first quadrant bounded by x = = 1, (x − 2)² + y² = 1, and x² + y² = 4.…

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Q: Evaluate the limit: r(t + h) – r(t) lim h→0 for r(t) = (1-6, sin t, 8) h r'(1) = ( ...

A: We have to find components of r't

Q: Let d(x.y) = |1/x - 1/y be a metric on R(0)XR\(0). Then the open ball of center 1 and radius 1/4 is*…

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Q: Let X = R², x, y EX where x = (x₁, x2), y = (y₁.Y2) with the metric function (4 pts) (x, y) =…

A: This is a question of topology and metric space.

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: Let X = {2,3, 4,..} and de fined metric on X as d (r, y) = |- yl %3D %3D Find орen ball B (2,2) =…

A: Follow the definition: B(2,2) = { x in X : d(x,2) &lt; 2 }