C is the intersection curve between the paraboloid z = x² + y? and the plane 6x + 8y – z = 21, oriented counterclockwise seen from above. Find the circulation 4x dr, (a) using Stokes' theorem,

Q: 1. Consider the ellipsoid x² + y² + 4z? = 12. (a) Find an equation of the tangent plane to the…

A: As per bartleby guideline for more than one question, only 1st one is to be answered. Please upload…

Q: 11. Use Stokes' Theorem to evaluate OF dr where F=(xz,e",y+z) on the curve that is the intersection…

A: The equation of curl is F=xz,ey,y+z. The equation of plane is 6x+3y+z=12

Q: The plane y=2 intersects the elliptic paraboloid ē =4x ty parabobo. Fied the slope ot the fangenh to…

A: Given :     y = 2      z = 4x2 + y2

Q: Evaluate fs F dS, where F(r, y, z) = (2², xy, z) and S is the part of the paraboloid z = 22 + y?…

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Q: Consider x=h(y,z) as a parametrized surface in the natural way. Write the equation of the tangent…

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Q: a) Evaluate x²y? ds, where the curve C is the intersection between the surface z = := V1– y? and the…

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Q: Find a vector parametric equation for the ellipse that lies on the plane 5y−3x+z=9 and is inside the…

A: Given that: 5y-3x+z=9 Let x=u cos v, y=u sin v So z=9+3x-5y  =9+3u cos v-5u sin v

Q: S is the part of the paraboloid z = x² +y? below the plane z = 4 oriented upward and F(x, y, z) :…

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Q: Find the flux of F (x, y, z) = 4.xi+ 4yj+ 2k outward (away from the z-axis) through the surface cut…

A: The field lines of a vector field F through surfaces with unit normal n, the angle from n to F is θ.…

Q: Find the parametrization r(t) = [x(t), y(t), z(t)] of the curve obtained by intersecting the…

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Q: Are there any points on the hyperboloid r2 - y? – 22 = 1 where the tangent plane is parallel to the…

A: Given hyperboloid ;    x2-y2-z2=1  and equation of plne which is parallel to tangent plane  x+y-z=0

Q: (b) Find a parameterization, in spherical coordiantes, for the ellipsoid 4x2 + 9y2 + 2²/4 = 1 that…

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Q: 2. Bounded by the paraboloid = 10 – 322 – 3y? and the plane z = 4 (using polar coordinates)

A: The paraboloid is bounded by z=10-3x2-3y2 . Now, converting in polar coordinates, z=10-3r2.

Q: Consider the curve C which is the intersection of the surfaces shown in the accompanying figure. X Z…

A: Consider the curve C:x+2y-8=0. Given figure:

Q: 2. Use Stokes theorem to evaluate fs da. a) a = x2 sin z da + y? dy + ry dz, S is the part of the…

A: As per our guidelines, we are supposed to answer only one question and rest can be reposted Given:…

Q: Use Stokes' Theorem to evaluate SSs curlF · dS, where F(x, y, z) = 7xyi + 3yzj+4z²k and S is the…

A: Consider the provided question, Use Stokes' Theorem to evaluate ∫∫Scurl F·dS,where F(x, y, z)…

Q: Consider x = h(y, z) as a parametrized surface in the natural way. Write the equation of the tangent…

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Q: z+v" = 2 +3: = 2

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Q: Let F = yz, xz, xy > . // curlF dS, where Use Stokes' Theorem to evaluate - x² – y? that lies above…

A: Given that F=-yz,xz,xy The paraboloid is z=8-x2-y2 and z=7 Finding the plane of intersection,…

Q: Consider the parametrization y: [0,47] → R, (t) = (t cos t, t sin t), of a counter-clockwise spiral…

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Q: (b) Give an example of a parametrisation of the unit circle, C = {(x, y) E R² | x² + y² = 1}, that…

A: The parameterization of a circle  that generates the clockwise orientation of C is given by: x=r…

Q: The surface 22 = x² +y? – 4x + 5 remains unchanged when rotated about the line L in R³ Which of the…

A: The given surface is z2=x2+y2-4x+5, which is as shown below: This is a hyperboloid paraboloid.…

Q: Calculate fc((3x + 5y) i + (3x + 4y) j) - dr where C is the circular path with center (a, b) and…

A: According to the greens theorem:

Q: H=pz coup ap te'sing a4 tp'ar At point (1, ,0)

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Q: Suppose that C is the curve of interesection between the plane z = x and the cylinder x² + y? be the…

A: Consider the given function. z=x and x2+y2=1 As, the function S be the interior of this ellipse…

Q: Show that the tangent plane to the quadric surface at the point (x0, y0, z0) can be written in the…

A: Given surface is x2a2+y2b2+z2c2=1 To find the equation of the plane, find the normal of the given…

Q: Let F = . Use Stokes' Theorem to evaluate curlF · dS , where S is the part of the paraboloid z = 14…

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Q: Let F (tanx + y)ĩ + (x + secy)j and C be the boundaries of the ellipse 9y=9 in the counterclockwise…

A: Given - F→ = tanx + yi^ + x + secyj^ and C be the boundaries of the ellipse x2 + 9y2 = 9 in the…

Q: Q2. Use the Stokes' Theorem to find fF.dr where F(x,y,z) = and C z=9-x²-y² on the xy plane oriented…

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Q: Integrate F =-(y sin z)i + (x sin z)j + (xy cos z)k around the circle cut from the sphere x2 + y2 +…

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Q: Find the circulation of F= (xy)i+(yz')j-(2xyz)k around the curve C in the counterclockwise direction…

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Q: Use Stokes' theorem to evaluate $ (xy – y) i – In z j+ (x +x²) k) • dr, where C is the intersection…

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Q: Find the work done by F in moving a particle once counterclockwise around the given curve = (2x –…

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Q: Compute e3z dz , Jy (z – 1)2(z+ i) where +y is any simple closed curve oriented counterclockwise…

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Q: a) Evaluate ds, where the curve C is the intersection between the surface z = V1- y? and the plane r…

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Q: where F(x, y, z) = (ry, x²y, y) Use Gauss's theorem to calculate where is the paraboloid z = 4 – x²…

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Q: Find the distance between two planes a and a1 that are tangent to the ellipsoid 2 +2y? + z² = 5/8…

A: We have to find the distance between the two tangent planes parallel to x+y+z=0  that are tangent to…

Q: b) Use Stokes' Theorem to evaluate fF•dr for F(x,y,z) =-yi+ xz j+ zk where C is the curve of…

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Q: Consider the ellipsoid V(x,y,z)=rx2+σy2+σ(z−2r)2=c>0. Determine a sufficient condition on c so…

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Q: Find D at P(0,0,8) due to a disk of surface charge 15nC/m2, p< 5 m lies on the z=0 plane

A: To find the electric field due to a charged disk. The surface charge of the disk is, σ=15…

Q: Use Stokes' Theorem to evaluate ∫CF⋅dr where F(x,y,z)=xi+yj+3(x2+y2)k and C is the boundary of the…

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Q: Use Stokes’ Theorem to evaluate F(x, y, z) = 3zi + 4x j + 2yk; C is the boundary of the paraboloid…

A: Use Stokes theorem

Q: Let F = . Use Stokes' Theorem to evaluate curlF · dS , where S S is the part of the paraboloid z = 9…

A: Find your answer below

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: a) Evaluate (-x i - yj+ 2z k) - n dS, where o is the portion of the paraboloid z = 3r2 + 3y? between…

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Q: Let F = · Use Stokes' Theorem to evaluate curi curlF · d§, where S is the part of the paraboloid z…

A: here i had written stokes theorem and done some basic calculation to find the curve of intersection…

Q: yz ds where o is the part of the plane x + y+ z = 1 that lies in the first octant.

A: Given The given integral ∫∫σyzds over the region x+y+z=1 that lies in the first octant can evaluated…

Q: Q: (a) C is the intersection curve between the paraboloid z = x² + y? and the plane 2.x – 4y + z =…

A: Consider the equation