(a) Prove that M: (x² + y^y° + 3z² = 1 is a surface. (b) For which values of e is M: z(z – 2) + xy = c a surface?
Q: 1. Let E be the solid under the paraboloid 2=r'+y. above the ry-plane, and inside the cylinder +y =…
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Q: Compute the flux of F = 3(x + z)i +j+ 3zk through the surface S given by y = x2 + z2, with 0 0, z 2…
A: flux is determined as shown below
Q: The surface area of the part of z = 2 + V4 – x² – y² that lies inside the paraboloid z x² + y² is
A: This is a problem of vector calculus.
Q: 5.Consider the ellipsoid V(x,y,z)=rx2+σy2+σ(z−2r)2=c>0.Vx,y,z=rx2+σy2+σz−2r2=c>0.…
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Q: - Graph the surface 1 = y² + z² — by first sketching at least three cross sections in each of the xy…
A: To Plot: Graph The Surface 1=y2+z2-x24 by plotting at least three cross section in xy and yz plane
Q: 2. The task is to determine the flux of B across G, where G is the positively-oriented portion of…
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Q: 14) Find the points on the surface xy- z2 = 1 that are closed to the origin. %3D
A: To find the point on the below surface that is closed to the origin. xy−z2=1
Q: 1. If F= (y,z,x), the outward flux through a smooth closed surface (a) depends on the surface. (b)…
A: Given, F=y,z,x We know, Flux by the Gauss divergence theorem. ∬F.ds=∬∫divF.dvF=yi+zj+xk If you…
Q: The attached figure shows the surface S1: y + z= 2, limited in the first octant with surface S2: x =…
A: The area of a surface z=f(x,y) bounded by the region R is given by the integral expression:-…
Q: Consider the solid E determined by the inequalities x² + y² + z≤ 4, and let S denote the surface…
A: “Since you have asked multiple question, we will solve the first question for you. If you want any…
Q: 16)Consider the equation of the surface S given by In (xyz) + xyz + x3 - y3 + z3 = 2 An equation for…
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Q: 2. Evaluate the triple in tegral where E is the y-2x2+2z20nd the plane y= 8. solid bounded by SI…
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Q: 1. Find the volume of the solid bounded by the paraboloids z = 6 – x² – y² and z = 2x² + 2y².Use…
A: From the given problem: z=6-x2-y2 ; z=2x2+2y2 For finding volume we using the triple…
Q: Example 3 a) Evaluate the surface integral SaS 2x²y dS over the surface y² + z² = 1 between %3D x =…
A: The given problem is related with surface integral. We have to evaluate the surface integral…
Q: What is the equation of the plane to the surface z = e3y sin(3x) +2 at the point P ,0, 3 O x - y -…
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Q: Find the points on the surface z2 = xy + 4 closest to the origin.
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Q: (4) Use Stokes' Theorem to evaluate 11. Vx (yz, xz, xy) dS where S is the surface S that is the part…
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Q: 3. Calculate the rotational flow of F(x, y, z) = -j + xj + x²k, that goes through the cone given by…
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Q: • SSS ²³² x² + y² = 2, and the xy plane. Evaluate e² dV where E is enclosed by the paraboloid z = 6…
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Q: 6. C is the curve of intersection of the plane 4x + 5y+ z = 0 with the cylinder x2 + y? = 4 directed…
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Q: 4. (Section 17.8) Use the Divergence Theorem to calculate [[F F-dS, where F = y + ryj – zk and S is…
A: To use the divergence theorem to calculate ∫∫SF⋅dS.
Q: 1. Consider the spiral ramp r(u, v) = ( u cos v, u sin v, v ) with 0 < u < 3 and 0<v < 2n. Use a…
A: Consider the given information.
Q: 2. For the surface z = In(y+ Vx² + y²). (b) Find the linear approximation at the point (3,-4).
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Q: 3. Evaluate the line integral y² dx + x²y dy where C' is the rectangle with vertices (0,0), (5,0),…
A: Solution is below
Q: The shape of a redwood tree (without any branches) can be modeled by an upside down paraboloid, f(x,…
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Q: 3. Evaluate /// 2* dV if G is a solid in the first octant bounded by the plane y + z = 2 and the…
A: triple integral
Q: 5) Evaluate the line integral S YX²dX + XY²dY on curve C, where C is the upper portion of the www m…
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Q: 4. Compute |F- dš, where F = (y, –x,2z) and S is the surface of the sphere of radius 2, centered at…
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Q: 9. Show that the sum of the T-, y-, and z- intercepts of any tangent plane to the surface VI + Vỹ +…
A: Given, Gradient of the surface at any point (x0,y0,z0) is given by,
Q: 11. Integrate G(x, y, z)=x√y² + 4 over the surface cut from the parabolic cylinder y² + 4z = 16 by…
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Q: What is the equation of the plane to the surface z = e3Y sin(3x) +2 at the point P,0, 3) O x- y + ez…
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Q: Solve the problem. 15) Find the centroid of the solid with constant density a enclosed between the…
A: Given :- The centroid of the solid with constant density α enclosed between the sphere of radius 2…
Q: The surface area of the surface generated by revolving y= 1-x2 for 0<x<1 about the x - axis is S = […
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Q: Sketch the trace curves of x^2 − y^2 + z^2 − 4x + 2z = 4, and use these trace curves to sketch the…
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Q: Find the maximum and minimum values of f(x, y, z) = x - z on the ellipsoid x² + 2y? + 22 = 1.
A: maximum=? minimum=?
Q: b) a dr+(x+yz) dy+(ry- VE) dz, C is the boundary of the part of the plane 3r+2y+z in the first…
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Q: Evaluate the triple integral. (x2 + y2) dx dy dz; W is the pyramid with top vertex at (0, 0, 2)…
A: The given triple integral is,
Q: Find the average height of the hemispherical surface z = √(a2 - x2 - y2) above the disk x2 + y2 ≤ a2…
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Q: Is it true or false that by using a suitable translation x = x' + xo, Y = y' + Yo, it is possible to…
A: Transforming the conic equation
Q: 6. Compute the flux of F = [0, 0, z²] through the spherical surface S given as the upper hemisphere…
A: To find- Compute the flux of F→ = 0, 0, z2 through the spherical surface S given as the upper…
Q: (b) Find the surface area of the portion of the paraboloid z-2-x -y above the xy- plane. AMINATION A…
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Q: 1. Let H be the hyperbolic paraboloid given by z = r² - y, and let P be the plane given by n x = 1,…
A: Given: H be the hyperbolic paraboloid z=x2-y2 and the plane P defined as n.x=1 where…
Q: 2. Suppose that the edge lengths r, y, and z of a closed rectangular box are changing at the…
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Q: Let the surface Σ be the solid bounded by the paraboloid z “ 1 ´ x2 ´ y2 and the xy´plane.…
A: Use Gauss divergence theorem to compute the outward flux.
Q: a) Evaluate the surface integral ſ 2x²y dS over the surface y² + z² = 1 betwee x = -1 and x = 5.
A: a We have to evaluate the surface integral ∬σ2x2ydSover the surface y2+z2=1 between x=-1 and x=5.…
Q: Find the points on the surface z²2=xy+1 that are closest to the origin using LaGrange multipliers
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Q: Let F = (x'e* +2.xy - 2y)i+(cos y+x² +3x)j and let C be the ellipse with equation (2x - 2y) +(x+ y)°…
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Q: (x - y) dv, where E is enclosed by the surfaces z = x² - 1, z = 1 - x², y = 0, and y = 4
A: Consider the given function fx,y,z=x-y. The region E is enclosed by the surfaces…
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- Consider the solid that lies above the square (in the xy-plane) R=[0,2]x[0,2] and below the elliptic paraboloid z=49-x^2-y^2Find the points on the surface x2 - zy = 4 closest to the origin.Use Green’s theorem to evaluate ∮C(ye2xy−5y)dx+ (xe2xy−2x)dy, where Cis the counterclockwise oriented boundary curve of the square with vertices at(0,0), (0,1), (1,0), and (1.1).
- 5.Consider the ellipsoid V(x,y,z)=rx2+σy2+σ(z−2r)2=c>0.Vx,y,z=rx2+σy2+σz−2r2=c>0. a.Calculate dVdtdVdt along trajectories of the Lorenz equations (1).1.Let D=6xy ax+ 2(xz+3z2)ay+3yz az C/m2.Evaluate the surface integrals to find the total charge enclosed in the rectangular parallelepiped 0 < x < 3, 0 < y < 2, 0 < z < 4.