Let X = R and x, y E R. Admit for all x and y the metric d(x, y) = |arctan x arctan y|. | Compare B(0, 2), B(0,4), B(1,4). What do you observe?

Q: 3. Use implicit differentiation to find əz - for ?х x²+2y²+5z²=2.

A:

Q: A circle is the set of points equidistant to a given center. Suppose distance is defined as d((x,y),…

A: Solution is written in below steps.

Q: E valuate f win Tydy tyxdx olE Cethe line seqment f&Om (0,2) to

A: on solving this we get the

Q: ds² = dt² – dz² – (1 + 2h(t − z))dx² − (1 – 2h(t – z))dy² - - how that this is a solution to the…

A: Let us see -  the metric given is a solution to the linearized empty space Einstein equations, we…

Q: The curve C is the circle of radius 5 contained in the plane z 5 and centered around the z-axis,…

A: Solution:-

Q: 11. Let F(x, y, z) = = (), and let P represent the path that follows the from (1, 0, 1) to (0, 1,0),…

A:

Q: Let F(x, y) = x²e*i+ 60x²y² j, and let C1 denote the curve consisting of the line segment from (0,0)…

A: Given, F(x,y)=x2ex,60x2y2 The curve C1 consists of the line segment Ca from 0,0 to 1,1, followed by…

Q: Problem 9 (6 points). For each of the following functions d: R2 x R² → [0, 00), say whether d is a…

A:

Q: et F(x, y) = (y sin x)i - 3x²j. et D be the unit square (i.e. D = {(x, y): 0≤x≤ 1,0 ≤ y ≤ 1}), and…

A: We have to choose the correct answers of two questions.

Q: Evalwate f, the inaicated point. (Y.y) and Fy y,y) CY.y) at 3. at (l,3)

A:

Q: The curvature of a function y = f(x) is given by the expression y" (1 + (y')²)³/2 Determine a…

A:

Q: f1(x) = sin(x1) cos(x2), T f2(x, y) = xy, f3(x) = xx¹, € R² x = x, yern XER"

A: Given the following functions                  f1(x)=sin(x1)cos(x2),   x∈ℝ2f2(x, y)=xTy,   x, y∈ℝn…

Q: Let F (x, y) = 8Q A. Compute 82 yi+zj z² + y² and let C be the circle r(t) = (cost)i + (sin t) j. 0…

A:

Q: convex function,not any other method

A:

Q: 3. Let C be an oriented curve parameterized by r(t), where a ≤t≤ b. Suppose that the following two…

A: The angle between F→and d→r is 0 degree as they are parallel to each other.

Q: 3: Let a function d: R* x R* → R defined by d(x, y) = E-\xi - yıl Vx = (x1,...,x4), y = (y1,...,y4)…

A: Here we will prove that d is a metric function.

Q: 1. Find all points on the graph of h(x, y) = plane is horizontal. 3x²9x + 4 + 2y³ - 12y where the…

A:

Q: Let F(x, y) == and let C be the circle r (t) = (cost)i + (sint)j, 0 ≤ t≤ 2π. A. Compute მი Note:…

A: Step 1: Step 2: Step 3: Step 4:

Q: Let F = (4x³ + yz, xz, xy). Let C be any smooth curve from (1, 1, 1) to (2, 3, 0) Calculate cF dr.

A:

Q: If F(x, y, z) = z - f(x, y) and s is the graph of z = f(x, y) and (a, b,c) is a point on the surface…

A:

Q: C is the part of the circle of radius 3 in the first quadrant from (3,0) to (0,3) and F = (1, -y),…

A: Given, C is part of circle of radius 3 in the first quadrant from (3,0) to (0,3) and F=[1,-y] The…

Q: Establish the interval where the function h ( x) *x° + 1 is concave down (CD), and find its point of…

A:

Q: Use Green's Theorem to evaluate | F. dr, where F = (Va + 5y, 2x + Vỹ ) and C consists of the arc of…

A:

Q: Let C be the curve connecting (0,0) to (1,2) to (0,6) to (0,0) using straight lines. (5x*y – 3x²y+…

A: Greens theorem based problem......

Q: Consider F and C below. F(x, y, z) = yz i + xz j + (xy + 12z) k C is the line segment from (3, 0,…

A: Introduction: To find the potential function for a given vector field first use the derivatives and…