3. (a) Let S be a compact regular surface. Show that there exists a point p on S such that K(p) > 0. (Hint: find a point p € S that maximizes the distance to the origin in R³. Show that K(p) > 0. You may find do Carmo, 1-5, problem 14 useful.) (b) Let S be a regular surface that is diffeomorphic to a torus. Show that S contains points where the Gauss curvature is positive, zero and negative.

Q: 2. State the names of the following quadric surfaces. (a) The surface z = y² – 2x². (b) The surface…

A: a. this equation is in form of  as sign of A and B is opposite it is  Hyperbolic paraboloid

Q: Q2 Find an equation for the surface that contains all such points P that has equal distance to the…

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Q: Let C be the positively oriented closed plane curve consisting of the line segment from (-3,0) to…

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Q: - Show that the equation of the circle passing through (1, 1) and the points of intersection of the…

A: we need to show that equation of circle passing through (1,1) and point of intersection of given two…

Q: 3. Let & be the surface with equation 2² 3- (y - 4√/2)² 4² - 1. 3² (a) Give equations of the traces…

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Q: sider the surface z = f(z, y)=√√/25 - 1² - y².

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Q: Describe in words the level curves of the top half of the sphere x2 + y2 + z2 = 1.

A: Top half of the sphere will be a hemisphere.

Q: Which of the following quadratic surfaces describe an elliptic paraboloid shape ? OA. z = 2? – y² +…

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Q: (х-2)? (у-3)? For the quadric surface, = z², the center of the cone can be found at which point? 9…

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Q: 1) The maximum distance from the origin of the xy plane to the circle described by (x – 12)? + (y –…

A: given equation of circlex-122+y-62=16 As per our guideline we supposed to answer only first question…

Q: Find the point on the sphere x+ (y-4) +(z-9) 9 nearest to a. the xy-plane. b. the point (0, -9,9).…

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Q: Name and graph the following quadratic surface. 9x2 + 4z2 – 36y = 0

A: For (b) Sketch of z=9-x2-y2: The graph is a paraboloid.

Q: (3) Consider the quadric surface SC R, having equation a2 + y? – 22 = 9. Find all lines through P =…

A: The quadratic equation is given by x2+y2-z2=9.  General equation of line is: x,y,z=a,b,c+tl,m,n…

Q: 6. In Section 12.5, you learned that the distance from a point (ro, Yo, zo) to a plane ax+by+cz = d…

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Q: Sketch the surface z = x² - y2 by slicing appropriately. Be sure to indicate the slice in at least…

A: Given: z=x2-y2 To do: Plot the given surface by slicing appropriately.

Q: (2) Let K be a 2-complex given in the following figure. B K : B. (a) Determine the surface which…

A: (a) The planar diagram by using the Euler's rule, Using the Euler rule, v-e+r=2 where v=5,e=8,r=5…

Q: Formulate an verify Green theorem when [P, Q] = [2yx, y² + 2x] and the curve y is the positively…

A: According to the given information it is required to formulate Green theorem. When

Q: Let (P) be the surface given by the following equation z = 5-x²-y² and (Q) be the surface given by…

A: Given surface P is, z=5-x2-y2    ----(1) For xy projection, put z=0 in (1) we get, 0=5-x2-y2⇒…

Q: 10. Consider a parabola y = x2 and a circle with center C(0,2), as shown below. The points A and B…

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Q: (P) is a circular paraboloid around z-axis and (Q) is the surface given by + += 1 This option (P) is…

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Q: 2. Consider the equation r = 4 – y². (a) Find a parameterization of the segment of this parabola…

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Q: [10] (2) GIVEN: The surface given by : x² + y² + 2z² = 6 and the line L: c(t) = (1-t, 2+t, 1+2t), t…

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Q: Find the point on the sphere x² + (y + 2)² + (z+ 9)² = 4 nearest to a. the xy-plane. b. the point…

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Q: How do I find the equation for the surface consisting of all points P for which the distance from P…

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Q: A rental truck costs $45.45 for a day plus 94¢ per mile. (a) If x is the number of miles driven,…

A: Given that, a rental truck costs $45.45 for a day plus 94c per mile.

Q: Let (P) be the surface given by the following equation + 4 and (Q) be a surface given by the below…

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Q: Suppose |||| = a, so that the tip of r is on a sphere of radius a with center at O. (a) Show that r…

A: Introduction: An orthogonal unit vector is a vector that is perpendicular to another vector and has…

Q: The level curve of the surface f(x,y) = x² + y2 + 4x -10y + 28 corresponding to c is a circle of…

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Q: (z-3√2)² Allow T to be the surface with equation 1 42 3² 1. Provide the equations of the traces of T…

A: here we use general equation of ellipse to conclude that the trace is elliptical.

Q: Let Ci be the line segment from the origin to (3, 2,0); C2 be the shorter circular arc of the circle…

A: The three curves given are as follows: C1: The line segment from 0,0,0 to 3,2,0. C2: The circular…