2. Prove that if d is a metric for X, then another metric d' for X is given by if a ≤ b " if b ≤ a (a) d'(x, y) a, b. (b) d'(x, y) = = min{1, d(x, y)}, where min{a, b} d(x,y) 1+d(x,y* = a b for any real numbers

Q: cost-sint) √2 r(t) = (3e' cost)i + (3e¹ sint)j + 2k For the space curve below, T= i+ sint+ cost s!).…

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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: Let k(x)=f Cx) A) K') B) K' (2) C)K' 3)

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Q: 2. Find the tangent plane to z = x f (t) where t = at a point (a,b) not on the y-axis. [Note: be…

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Q: 3. Let C be the circle x + y = 16 and let S be any surface above the xy-plane with C as its…

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Q: Prove that the parabola y = x^2 at the points where the line has an angle of inclination "o" the…

A: Curvature of  a graph y= f(x)  k=d2ydx21+dydx232

Q: 5. Let C be the circle x +y} =4_which is positively oriented. Let C, be the upper half-circle C1 =…

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

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Q: 2. Draw a contour map of function f(x, y) = 2x/(x² + y²) showing several level curves.

A: We Know that  The level curves of a function f of two variables are the curves with equations f(x,y)…

Q: Let f(x.y) = ry. a. Sketch/describe the domain of g Domain = { (x,y)]yz0} b. Sketch a contour map…

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Q: Let S = R^2 - {(0, 0)} and let L be the set of all Cartesian lines lying in S. Show that {S,L} is…

A: We will use the definition of incidence geometry to solve the problem.By definition, a geometry…

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 AABC be a triangle S². Let the dual point C'ES² of C be defined by the following three…

A: Given: △ABC is a triangle S2 (the two-dimensional sphere). The dual point C'∈S2 of C is described by…

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

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

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Q: =

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Q: 19. If F(2, y, z) = (2xz, 3y - 22,1 + a2) and C is the path along a2 + y? 1 in the plane z = 0 from…

A: To find: Work done by the field on the particle moving along the loop.

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: Circle the equation that generates the surface shown below.

A: Solution for 9 (a): The equation that generates the given graph is choice (e): x2+y22=z2

Q: For the first few problems in this set, let (3) 1 F₁ = 0 F₂: = 1. Let S denote the unit sphere cos u…

A: Given a parametrization of the unit sphere, ST→(u,v)=(cos⁡usin⁡vsin⁡usin⁡vcos⁡v)Partial…