Let S be the surface parametrized by R (u, v) =< u, uv, uv

Q: Describe the surface r(u, v) = ⟨2 cos u, 2 sin u, v⟩, for 0 ≤ u ≤ π and 0 ≤ v ≤ 1.

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Q: Let S be the portion of the cylinder y = ln x in the first octant whose projection parallel to the…

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Q: What is the normal vector, n for S is a surface of x +y² +z² = 36 in the first octant? %3D

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Q: Describe the set in cylindrical coordinates. y^2 + z^2 ≤ 9, x ≥ y

A: Let x=rcosθ, y=rsinθ  and z=z. And the given equation is x ≥ y ⇒ cosθ ≥ sinθ

Q: Find the tangent plane and normal line to the surface f (r, y) = In (2r + y) at the point (-1,3,0).…

A: “Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: Find a unit vector normal to the surface x^3- xyz + z^3 = 1 , at the point (1, 1, 1)

A: To find the unit normal vector to the surface x3-xyz+z3=1 at the point 1, 1, 1

Q: What is the value of F.ds, where F = 4xz,- y° i, + yz, ? Here, s is the surface bounded by x = 0, x…

A: Here function is given find f.ds as below

Q: Use Exercise 22 to find the centroid of the triangle with vertices (0,0), (a, 0) and (a, b) where…

A: To find the centroid we add the x coordinates and the y coordinates respectively and then divide by…

Q: Evaluate the surface integral xyzdS where S is the part of the plane z = 1 + 2x + 3y that lies above…

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Q: Compute the surface area of the surface segment parametrised by x(u, v) = u² cos v, y(u, v) = u² sin…

A: Given, xu,v=u2cosv,yu,v=u2sinv,zu,v=u2where 1≤u≤3 and 0≤v≤2π To find the surface area.

Q: Sketch the surface z = f(x,y). f(x, y) = -x 2 - y 2 USE GEOGEBRA

A: Function f(x, y) = -x2-y2 To Sketch the surface z=f(x, y).

Q: Find the area of the surface given by  r(u,v) = (u+v,u,v) 0 ≤ u ≤ 1, 0 ≤ v ≤ 1

A: Given ru,v=u+v, u, v0≤u≤1 , 0≤v≤1

Q: Calculate the normal vector, n for the surface z=5-x-y with upward orientation.

A: We have to find the normal vector.

Q: Let S be the disk enclosed by the curve C: r(t) = (cos cos t, sint, sin p cos t), for 0sts2r, where…

A: We need to find circulation.

Q: Evaluate Curlv-ñ, where v = 2xyi + (x-2y)j+xzk and ñ is a unit vector normal to the surface shown in…

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Q: Find the normal vector to the surface z = xe™Y at the point (2,0, 2).

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Q: Let C be the curve intersection of the sphere x'+y+=9 and the cylinderx+y° =5 above the xy-plane,…

A: To Evaluate: ∫CF·dr Using Stoke's theorem. where F→=&lt;2yz,5xz,lnz&gt; and C is the curve of the…

Q: What is the value of J]F ds, ds, where F = 4xzi,- y² % + yzi, ? Here, s is the surface bounded by x…

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Q: Use Stokes' Theorem to find the line integral of F = 3yi + 2zj + 5xk around a 5, oriented…

A: The solution are next step

Q: Evaluate Curlv•ñ , where v = 2xyi + (x² – 2 y)] + xzk and ñ is a unit vector normal to the surface…

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Q: Exercise 1: Evaluate the line integral xyzds where C is parametrized by i = 2ti +3sin t j+3costk;…

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Q: Describe the surface r(u, v) = ⟨4 sin u cos v, 4 sin u sin v, 4 cos u⟩, for 0 ≤ u ≤ π/2 and 0 ≤ v ≤…

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Q: Find a vector parametrization for the intersection of the plane x + y+z = 3 and the cylinder x² + z2…

A: solution given below..

Q: Assume S is a smooth surface parametrized by R(u,v) = (V / za sin y do. Evaluate S Hint: Find the…

A: We use double integral to solve this.

Q: Find a parametrization of the curve. The intersection of the surfaces z = x^2 − y^2 and z = x^2 + xy…

A: Given : The surfaces z = x2 - y2 and z = x2 + xy - 1.   To find the parameterization of the curve.

Q: Find the surface area of the helicoid (spiral ramp) with vector equation r(u, v) = (u coS v, u sin…

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Q: A region R consists of a square bounded by the lines x = −7, x = 7, y = 0, and y = −14and a half…

A: Draw the region R as follows.

Q: Compute F - dS for F = (4e*, 2z, 2x), S being a surface with parametrization (r, s) = (rs, r + 4s,…

A: Given F=4ez,2z,2x, S being a surface with parametrization φ(r,s)=rs,r+4s,r, 0≤r≤1, 0≤s≤3, oriented…

Q: = xyz exists over a surface S c ed by the plane x-0,y%3D0,z%3D0, z=y V dS over the curve surface.

A: Given:  V= xyz x2+y2=9 To find: ∫s V ds  For solving the convergence x2+y2=9 (mod 1s) we use Chinese…

Q: Find the area of the surface. The helicoid (or spiral ramp) with vector equation r(u, v) = u cos(v)…

A: Follow step 2nd .

Q: Evaluate the line integral, where C is the given space curve.     Cxeyz ds, C is the line segment…

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Q: Find a vector normal to the surface x2 + yz = 5 at (2, 1, 1).

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Q: (a) Let S be the surface z2 = x² + y? bounded between the planes z = 0 and z = 2. Evaluate (z –…

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Q: Integrate G(x, y, z) = xyz over the surface of the rectangular solid bounded by the planes x = ≠a, y…

A: When we consider the face we find Using the formula of we have

Q: Find the tangent plane at the point (1,-1,1) of the surface f(x,y,z)= x^2y+y^2z+z^2x. %3D

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Q: A parametrization of the portion of a surface z=x²+y2 below the plane z= 16 is given by 7(u,v)=…

A: Since you have sent more than one question so we can just answer one of them because of our…

Q: Give an arclength parametrization of the line which is the intersection of the tangent planes of…

A: Let First, find the tangent planes at (1, -1, 0), (1, 2, 9).     Tangent plane at (1, -1, 0):…

Q: Sketch the section of the surface at z = 0 on y = sinx in xy-plane on [0,27). y = sinx 4 56 1 3. 6…

A: The given equation of the curve is: y=sinx To draw the given equation on the x-y plane we need to…

Q: 2. Calculate the centroid of the thin plate R which bounded by r = 1, y = 0 and, 2² + y² = 4.

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Q: Let S be a square plate with boundary C, having each side of length 2 units, lying in the plane x =…

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Q: Let S be a square plate with boundary C having length of each side as 2 units, lying in the plane z…

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Q: Let F(r, y) = 4xî + 3yî and let i be the outward unit normal vector to the positively oriented…

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Q: Find the surface area of the helicoid (spiral ramp) with vector equation F(u, v) = (u cos v, u sin…

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Q: the tangent plane of the surface [ z=x^5+e^(xy)cosyx] at the point (1,0) O z=5x-y-3 O z=5x+y-3 O…

A: Above question is solved.

Q: Use Stokes' Theorem to find the circulation of F = 4yi + 2zj + 4xk around the triangle obtained by…

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Q: Find the area of the surface. The helicoid (or spiral ramp) with vector equation r(u, v) = u cos(v)…

A: We have to find the surface area from the given condition

Q: 9. Let S be the surface parametrized by (u, v³, u + v) where 0 <u <1 and 0 < v<1. Calculate y dS.

A: The solution are next step

Q: Sketch the section of the surface at z = 0 on y = sinx in xy-plane on [0,2x]. y = sinx 1 3 4 5 1 1 2…

A: Let's find.

Q: Verify Stoke's theorem for F = -2yzi + vj +3xk over the surface defined z = 5- x² -y? above the z=1…

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Q: 2)Use Stokes' Theorem to evaluate | F.dr where F =2xyi- yj+2xzk and S is the first octant surface of…

A: The given problem is to evaluate the line integral in the surface of first octant with upward…