Let (X, d) be metric and PXIR space, dosy) I+ dexy) a de fined by qusN pusy) = ニ show that p is metric.

Q: find flux f Use divergence Be 1ector F-12z+2リズ分カーズド cut of Sphere *+jr* =9 theorem to 2 A ースK

A: given vector F→=12x i^+2yz j^-z2 k^ out of sphere x2+y2+z2=9 claim- use divergence theorem to find…

Q: Plane R is tangent to sphere 0 at point T. How are radius OT and plane R related?

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Q: Does the Divergence Theorem apply to surfaces that are not closed?

A: By Stokes' theorem:

Q: ,A, B,C be 4 points in R², no three of which colinear. Fix A' e VA, B' e VB and C' e VC h that AB ||…

A: In the following question, concept of parallel line is applied. Parallel Line Parallel lines are…

Q: Let S be the paraboloid y = x^2 + z^2. Find a parametric representation for S and its tangent plane…

A: The given data is: Let S be the paraboloid y=x2+z2 Find a parametric representation for S and its…

Q: The metric for a three-sphere is ds² = R² (d² + sin² § (d0² + sin² 0d²)) where ranges from 0 to 27,…

A: We are given that the metric for a three-sphere is ds2=R2dξ2+sin2ξdθ2+sin2θdϕ2 where ϕ ranges from 0…

Q: Consider 2-dimensional hyperbolic space with metric ds ds'+dy and y > 0. The length of a line…

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Q: Find the curvature of ƒ(x) = ln cos x, < x < 2 -- 2°

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Q: Let (X, d) be a metric space, and p: X→ IR doxy) I+desy) de fined by puM pusy) = %3D show that p is…

A: Consider a set X and a function defined from it to the real numbers as d : X×X→ℝ. Then, X,d is a…

Q: If N is a sphere of radius R that is centered around the origin. Use appropriate functions and…

A: Simplify the integral:  ∫Ωr→r3∇→r4-R4r5+7r3+2dτ=∫Ωrr3r^r^ddrr4-R4r5+7r3+2r2sinθdrdθdϕ…

Q: A space has the line element, ds? = v*du? +u*dv² . Determine whether this space is flat or curved.

A: It is given that the line element ds2=v2du2+u2dv2. it is required to determine that whether this…

Q: 4. Let r(s) be a unit speed curve with a curvature of zeroc fo. prove that the image r lies in a…

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Q: Let S be the upper half of the unit sphere x +y +=1 and take n as the upper unit normal. Use Stokes'…

A: Stokes theorem is used to find the line integration

Q: 3) use implict differentiation to finol dd et the point (8.1). 4 xy+ slny=32 Anywery: 28 a) 37 り 36…

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Q: Let = (0,0), and a = (2,-1) be points in R². Set G=B¹2 (0,1)={v = (x, y) = R²: d₂(0,v) <1} H = B¹¹…

A: Let, and  be points in  SetTo find:Sketch  and The coordinates system,, and  be points in  Set

Q: Integrate h(x, y) = (x+y)i + (y²-x)j over the closed curve that begins at (-7,0), goes along the…

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Q: 3. Let d₂(x, y) = |arctan(x) — arctan(y)| for x, y € R. (a) Show that d₂ is a metric on R. (b) Are…

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Q: 1. Vector Functions and Space Curves a) Find the domain of the vector function: r(t) = (√4 — t²,…

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Q: 4) Find the families of lines of curvature of the surface z = r + y? .

A: Given: z=x2+y2

Q: FG)= St SnA AS Crtrial purnts and at the 1.E va luate endponts of the gruen inteval Iden tfn (ocal…

A: Given: f(x)=x+sin x  (0≤x≤2π)To find the critical points, endpoints, local/global maxima and minima…

Q: Let C be the line segment from point (0, 1, 1) to point (2, 2, 3). Evaluate line integral yds. C

A: To evaluate the line integral

Q: 5. Let X = R? and define d, : R? ×R² →R by d (x,,y,),(x,,y2)) = max {x, – x,|-\y, -y}} . 13 a)…

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Q: Consider the function f( + ty)- 3z²y+i ain z nw that thin funeticon does not satisfy the…

A: a) The given function is: fx+iy=3x2y+isin x Cauchy Riemann equation:  Let fz=u+iv be a complex…

Q: (d) Let (Fn) be a sequence of closed subsets of M, and S, (j = 1,...,5) be subsets of M such that.…

A: We are authorized to answer one question at a time, since you have not mentioned which question you…

Q: 2. Find the limit of the following vector functions: а) lim (2eti + 6e-tj – 4e-2tk) t→ln 2 b) lim i…

A: Logarithmic and exponential properties,Differentiation formulas,l'hospital's rule,

Q: 5. Express the flowing quantities in rectangular form. a) sec( + j1) cot(17/12 + j3 cosh(2 + j3) b)…

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Q: A secant to two congruent circles, which is parallel to the line of centers OO', meets the first…

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Q: Find T, N, and k for the space curve r(t) = (7e'cos t)i + (7e'sin t)j + 4k.

A: Given data:  r(t)=(7et cost) i +(7et sin t) j+4k