Q.Assume that a surface (S) has the fundamental forms coefficients E =1 and F= uv, G 1, L= u, M u+ v, N 1. Then the surface (S) has 1.Many umbilical points. 2.One umbilical point. 3.Two umbilical points. 4.No umbilical points. 5. The given information is not enough. |3| Option 1 Option 2 O Option 3 Option 4 Option 5

Q: 5. Consider the surface: x(u, v) y(u, v) z(u, v) where u € [0, 1] and v € [0,2]. Calculate the area…

A: Introduction: The area is the area occupied by a two-dimensional flat surface. It has a square unit…

Q: Suppose S is a smooth surface parametrized by r(u, v) over D = [0, 1] x [0, 1] with ru = and rv = .…

A:

Q: Calculate s f(x, y, z) dS for the given surface and function. z = 5 – x? - y², 1<z<4 x² f(x, y, z) =…

A: According to the given information, it is required to find the integral for the given surface and…

Q: 4. calculate JI, † (x, y, z) dS for the given surface and function. x² + y² + z² = 4, 0 < y < 1; f…

A:

Q: Find the extreme values of ƒ(x, y, z) = x - y + z on the unit sphere x2 + y2 + z2 = 1.

A:

Q: 4) The half-widths of a ship with a length of 60 m at the loaded waterline are given in the table…

A: As interval is divided into 6 equal parts . So, n=6 h=x1-x0=1-0=1 y0=f0=fx0=f0=2.85…

Q: What is the equation of the surface revolved about the x-axis for z² - xz = ln x + 3x A. y² + z² =…

A: Given: The equation z2-xz=lnx+3x which is of surface and is quadratic for z.  To find: The equation…

Q: 7 Set & be the surface R gwen by in X = uV where Isusa y =y3 Evalvate: J Edxady +xdy adz

A:

Q: The volume in the first octant bounded by the surface x = 1 and x2 = y + 2z.   Group of answer…

A:

Q: Consider the surface shown in the figure below: Let h, k, r, m, n, t ER such that m, n, t>0. A…

A:

Q: e surface generated by y=x^3 on [0,1], about the x-axis. Final Answer is A(s)=3.563 sq. units, Show…

A: Solution:- y=x3       0,1 y'=3x2 y'2=3x22y'2=9x4 Now find surface area S= ∫012πy1+y'2dx…

Q: 10. Find all the planar and hyperbolic points on the surface o(u, v) = u-3uv².

A: A point on a surface is said to be a hyperbolic point if the Gaussian curvature at p,i.e., κp&lt;0.…

Q: Calculate s f(x, y, z) dS for the given surface and function. y = 1 – z?, 0 < x < 2, 0< z < 1 f(x,…

A:

Q: 4. Show that the set of points (z, w) e C with w2 = sin z is naturally a Riemann surface. %3D

A: Let, S=z,w∈ℂ2|w2=sinz be the set in equation. The multivalued analytic function fz=sinz can be…

Q: 17. Find the exact area of the surface z = 1 + 2x + 3y + 4y2, 1x 4, 0 y < 1

A:

Q: 9. What is the xy trace of the sphere (x – 5)2 + (y – 2)² + (z – 10)² = 16 ? а. (х — 5)2 + (у — 2)2…

A: To find the xy trace of sphere, put z=0 in equation of sphere.

Q: a. Which of the following surface graphs represents f(x,y) = y²? OA.

A: Given, the surface                           z=f(x, y)=y2

Q: 3. Evaluate , (x + z)dS where the surface S is the first-octant portion of the cylinder y² + z² = 9…

A: Step:- 1 Since, we know Surface integrals ∫∫Sf(x,y,z)dS=∫∫Df(x,θ) rx×rθ dxdθ…

Q: surface is γ(x,y)=(x,y, √(4−x2−y2) with 0≤ x2+y2 ≤ 2. 2. The parameterization of the surface is…

A:

Q: Describe in words the region in R' represented by the equation x² + y² – 10y + z² + 2z = 0 O A…

A: we have to check the region of the given equation in R^3 is

Q: Consider the surface S: z = 4 - x2 , in the first octant and limited by x2 + y2 , such and as shown…

A: The equation of the surface is given below. z=4−x2 The surface S is the part of the above surface in…

Q: 12) Consider the surface: S: z = 4– x2, in the first octant and bounded by x? +y², as shown in the…

A:

Q: Find the exact area of the surface z = 1 + 2x + 3y + 4y2, 1 ≤ x ≤ 11, 0 ≤ y

A:

Q: 3) . Find the centroid of the upper half of a sphere of radius 1, that is, of the set of points in…

A: Given the upper half of the sphere with radius 1, that is x2+y2+z2=1,z≥0. Find the centroid x¯,y¯,z¯…

Q: Find the area of the following surface. z=2/3(x^(3/2)+y^(3/2)), 0<=x<=1, 0<=y<=1

A:

Q: (a) Two surfaces are called orthogonal at a point of intersection if their normal lines are…

A: To show that two vectors are orthogonal, show that the dot product is zero.

Q: 4. Given an example of a change of coordinates z= r(s,t), y = y(s,t) where every region in the st…

A:

Q: 1. Let a, b, c ER and let Sa,b,c = {(x, y, z): ax2 + by2 + cz²=0, x² + y² +2²=1}- (1a) For which…

A: Given: Sabc=(x,y,z): ax2+by2+cz2=0, x2+y2+z2=1 To find: the values a, b, c for which Sabc a regular…

Q: 16) Consider the oriented curve C, boundary of a region D, given by C = C₁ UC₂, where C₁: y=x² y C₂:…

A: Given that C:C1∪C2 is a closed curve. C1:y = x2C2: y = 8 - x2 Find the another form of the integral…

Q: 10) Use Green's Theorem to eualuate S+10ny +y?)dx + l6xy+ 5a²) dy Toxy+ sa?) dy where C is the…

A:

Q: 3.4.8 Show that the sphere x² + y² + z² = 1, z ≥ 0. JJ,z²dS=, where S is the surface of

A:

Q: 10/ Compute S Fids, uhere S is the surface oriented up ward and E

A: According to divergence theorem ∫∫SF.ds=∬∫Vdiv F dxdydz Given F=y, -x,…

Q: 10. Using Stoke's theorem, evaluate y dx+ xz dy+x² dz where C is the boundary of the triangle cut…

A:

Q: Assume that [a, b] is a closed interval contained in the open interval (-1, 1). Let f(x) = V1 – x²,…

A:

Q: Evaluate the surface integral. ∬Sz2dS,S∬Sz2dS,S is the part of the paraboloid x=y2+z2x=y2+z2 given…

A: Evaluate the surface integral. ∬Sz2dS,Sis the part of the paraboloid x=y2+z2 given by 0≤x≤4

Q: ind the area of the surface generated when the give curve is revolve about the x-axis    y=(x^3)…

A: Consider the given curve: Here the objective is to determine the area of the surface generated when…

Q: Consider a surface S:Z = (x-1)'+2 in the first octant limited by y=-4x(x-2) sShawn ju the figere. z…

A: We will construct the appropriate surface integral to solve  the question

Q: 4. Suppose that VxF= (0,1,-1), S is the plane x + y + z = 1 in the first octant and Cis the boundary…

A: Given: ∇×F=01-1 S is the plane x+y+z=1 in the first octane  and C is the boundary of S. Now, to find…

Q: Let h, k, m, neR such that n > m >0. A general equation corresponding to the above surface is: A) (z…

A:

Q: Consider the surface S given by: S :r(u, v) = (uv², u + v, v³), u E [0, 1], v E [0, 2]

A:

Q: 15.4.10. ſſy(x + y + z) do, M is the surface x? + y² + z² = 4, z > 0.

A: We need to find surface integral.

Q: Given that o is the part of the surface z = 4 – x in

A: Given, the surface z = 4 - x in the first octant from y = 0  to  y = 2. If ψx, y, z=xyz, F(x, y, z)…

Q: 7. D is bounded by the surface (a2 + y² + z²)² = a²(x² + y? – z²). %3D

A: Sol

Q: 15. The solid enclosed by the sphere x2+ y2 + z2 = 4a? and the planes z = 0 and z = a. %3D

A: Given: x2+y2+z2=4a2          ... (1) z=0z=a

Q: 5. Prove that the surface E, = {(x, y, z)|a² + y² = 4} is locally isometric to the surface E2 = {(x,…

A:

Q: 15 - Consider the surface defined by z = f(z, y) = x³ + 2x²y +y². Find the equation of a normal line…

A:

Q: For the surface given by z= f(x,y) = 1 - 6z + y-3y? %3D %3D which of the following is true. Select…

A:

Q: (b) A = 2rza, + 3z sin ø a, – 4 rcos o a̟ and S is the surface of the wedge 0 <r< 2, 0 < 0 < 45°, 0…

A: