i) Pe(x) = 0 iff r € Ë. ii) Prove that pe is uniformly continuous.

Q: „Consider the space (R,T). where t={R,Ø}U{ACR:[0,1)CA} :Then (R,T) is not T, O T, but not T, O T2 O…

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Q: Let d be metric on X, and de f ined di on X as di = max (d (r, ).0}, then dis semi metric but not…

A: Given d be an metric on X and d1 is defined on X by  d1=maxdx,y,0                     (i) To verify…

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: Prove that the volume of the sphere x² + y² + z² = a² is V =÷na³

A: In this question, we find the volume of sphere by using triple integral

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

A: Given vector field is  Divergence theorem

Q: 1. Let d, be a metries on Xo and let the function a: X x XR defined be dr. y) dis. y)= prove that…

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Q: Assume s is a smooth surpace parametrized by e (u,v): <Jv, win' u, SZ-v ). Evaluatk 2x sin y do.

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Q: Let I = , zdV ,where D is bounded by the planes x = 0, %3D 6. z 0, x+y = 2 and the cylinder x2 + z2…

A: Given: I=∫∫∫D z dVwhere D is bounded by the planes x=0, z=0, x+y=2 and the cylinder x2+z2=1 in the…

Q: Let d: R x R R. defined as d (z, y) = |#1 - ml wherez= (21,2), y= (y.), then d is %3D Osemi metric…

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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: Suppose that the domain of the function f(z) = x2+2i is the closed unit disk D={z: \z|1} on the…

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Q: Let F= [x, y, z] and S be a sphere of radius R. Using the divergence theorem, the value of the…

A: Given : F = x,y,z

Q: Use Divergence Theorem to evaluate .(v²z²i +z²x²j+x²y²k).ds, where S is the upper part of the sphere…

A: Gauss Divergence Theorem : ∫∫SF·ndS=∫∫∫VdivF dV From ∫∫Sy2z2i+z2x2j+x2y2k·dS, F=y2z2i+z2x2j+x2y2k

Q: Verify the divergence theorem, ffaw F · dS = S[fw, Div FdV, for F(x, y, z) = (z)i + (y)j + (x)k,…

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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: 3. Suppose f(z) is continuous in the closed disk {려くR) and analytic in the open disk {|z| < R). Show…

A: To extend Cauchy's theorem to the boundary of the domain of anaylyticity ,under the given conditions

Q: Suppose f(x, y) satisfies the basic existence and uniqueness theorem in some rectangular region Rof…

A: Consider the given: Let the function f(x,y) satisfies the basic existence and uniqueness theorem in…

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: Let I = [[, ydV ,where D is bounded by the planes y = 0, z = 0 , x+ y = 5 and the cylinder y2 + z2 =…

A: We have to solve given problem:

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: Use Divergence Theorem to evaluate the flux integral ∫∫F·ndS, where F = 〈x^2, y^2, z^2〉, S is the…

A: Divergence theorem:∫∫sF.⇀  N^dS =∬∫DdivF⇀dV , here D is the solid bounded by the closed surface S.

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: State what the Divergence Theorem says, using your own words. You may assume that E is a solid in R³…

A: We can state the Divergence theorem .

Q: 5. Compute / | F. aš, where F = (y, -x, 2z) and S is the surface of the sphere of radius 2, centered…

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Q: Evaluate the line integral, where C is the given space curve.     Cxeyz ds, C is the line segment…

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Q: Calculate ( (2x + y) dV , where V is a closed space bounded by the cylinder z = 4 – x? and the…

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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: Find the volume of the Intersection curve C C Projection of C on ay-plane

A: Given: Given paraboloids are z=x2+y2 and z=8-x2-y2 We want to find the volume bounded by the…

Q: Use Divergence Theorem to evaluate .v²z²i +z²x²j+x²y²k).ds, where S is the upper part of the sphere…

A: To evaluate the below integral. ∫∫sy2z2i+z2x2j+x2y2kds¯

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

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Q: Show the equivalence of the three definitions for the following 2-dimensional manifold: implicit…

A: Given: The given statement is shown as follows, implicit surface x2+y2=z2.

Q: Use the Divergence Theorem to evaluate |F.n do where F(x,y,z)=(x² +cos yz, y – e" ,z² +x*) and S is…

A: According to question given :F(x,y,z)=x2+cos yz, y-ez, z2+x2S is the boundary of the solidBounded by…

Q: Evaluate both sides of divergence theorem for A= pa, + pcos(6)a+za, through the closed surface 0<p<…

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Q: Use the Divergence Theorem to evaluate (4x + 7y + z²) dS where S is the sphere x2 + y2 + z² = 1.

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Q: An _____ is a set of points (x, y) in a plane such that the sum of the distances between (x, y) and…

A: The sum of the distances to any point on the ellipse (x,y) from the two foci (c,0) and (-c,0) is a…

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

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Q: Let d : R x R → R be defined to be d(x, y) = | arctan(x) – arctan(y)|. Show that d is a metric on R.

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Q: Use the Divergence Theorem to evaluate (4x + 3y + z) dS where S is the sphere x2 + y2 + z2 = 1.

A: The divergence theorem establishes the equality between surface integral and volume integral.

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

A: Given problem is :

Q: for x=(xy xn) € R", define the li -ToRm by ||selle₁ = ½ |xil lacil || i=1 Rove that is actually a…

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Q: Verify Divergence theorem for the function F = e,p over the semi-cylindrical volume { x* +y° sa…

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Q: Let S be the sphere of radius 2, oriented inward. Let F = (2x + e", 4y + e*, 2:2). Use the…

A: In the given question we have to solve the double integral s.t∫∫F.ds  to the help of divergence…

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: . 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: Verify Divergence Theorem for F = (x+ y²) î – 2 xj + 2 yzk and the volume of a tetrahedron bounded…

A: Given that F=(x+y2)i-2xj+2yzk Take x=0, y=6-2z, Then,…

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.