Let F [e, e2, e*]and C be the boundary of the surface S: z=x², 0

Q: A parametric representation of the surface y? + z² – x² = 0 in R³ is given by, Select one: O a. æ =…

A: option (b) is correct.

Q: Let G(x, y, z)= (1+9xy)2 and consider a surface S given by the parametrization r(u, v) = [u, u³, v],…

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Q: (5) Use Stokes' Theorem to find the line integral |(* + 2y°)i + (y + z°) +(= + 2r*)k) • dř where C…

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Q: Let E be the tetrahedron in space with vertices (0,0, 0), (1,0, 0), (0, 1,0), and (0,0, 1). Let F(x,…

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Q: Verify Stokes' Theorem for F = xyî + yzĵ + xzk and where S is the surface of the paraboloid z = x²…

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Q: Let G(x, y, z)= (1+9xy)2 and consider a surface S given by the parametrization r(u, v)= [u, u, v],…

A: Step:-1 Given that Gx, y, z =1+9xy12  And   r u, v = u, u3, v0≤u≤1,  -1≤v≤1 So,  Gx, y, z =1+9xy12…

Q: Let S be the parametric surface, parameterized by r(u, v) = e" cos vi + e" sin vj + uk, (u,v) E D =…

A: given S be the parametric surface, parametrized by  ru,v=eu cosv i^+eu sinv j^+u k^  here…

Q: Evaluate the surface integral Il. f(z, y, 2)dS, where f(x, y, 2) = /1+ æ² + y? and S is parametrized…

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Q: Evaluate the surface integral I F. dS when F(x, y, 2) = y²i+ 3x² j and S is the graph of - ža (1+æ²…

A: Here, Fx,y,z=y2i+3x2jz=121+x2-y2

Q: Let F(z, y, 2) = (y, Y, z + z²), and let S'be the surface of a very small sphere centered at (-1,0,…

A: The sphere has the equation (x+1)2+y2+(z-1)2=r2, where r is the radius of sphere. If the sphere is…

Q: Use Stokes' Theorem to find the line integral [((x² + 2y² ((x + 2y²)i + (y + 2²)j + (2 + 2x²)k) · dř…

A: Given that C is boundary of triangle with vertices 1,0,0,0,1,0,0,0,1 So, equation of the plane with…

Q: Use Stokes' Theorem to find the line integral [((x² + x + 2y²)i + (y + z³)j + (z + 2x²)k) · dr where…

A: To Evaluate: The integral ∫C(x+2y2)i+(y+z2)j+(z+2x2)k·dr. where C is the boundary of the triangle T…

Q: Let E be the surface given by the parameterization r(6,0) = 3 sin() cos(0)i+3 sin() sin(0)j +3…

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Q: Given that σ is the part of the surface z = 2 − y in the first octant from x = 0 to x = 1. If ψ(x,…

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Q: Use Stokes' Theorem to evaluate F. dr where F(x, y, z) = xi + yj +Xx² + y² )k and C is the boundary…

A: Stokes theorem: ∮f→·dr→=∬S curl f→·n^ ds Given F(x, y, z)=xi + yj+9(x2+y2)k and C is the boundary of…

Q: Evaluate the surface integral G (x, y, z) dS C where G (x, y, z) = x – y – z, and C is the…

A: Given: Gx,y,z=x-y-zRu,v=u-v,v-uu-3v-10≤u≤20≤v≤3 Find: ∫∫CGx,y,zdS Formula used: ∫xndx=xn+1n+1∫dx=x

Q: Let F [e', e', e*] and C be the boundary of the surface S. z=x, 0sx<1,0<y<1. If we use the theorem…

A: Given F^=ey, ez, ex and C is the boundary of the surface S: z=x2,0≤x≤1,0≤y≤1. The stokes theorem is…

Q: Use Stokes' theorem to evaluate ∮C z2ex^2 dx + xy2dy + tan− 1 (y)dz where C is the circumference x2…

A: Application of Stokes Theorem:

Q: Attached picture is the question

A: Consider the provided question,

Q: Use Stokes’ Theorem to evaluate ∫c F ⋅ dr. In each case C is oriented counterclockwise as viewed…

A: Given: f(x,y,z) = xyi + yzj + zxk C is part of paraboloid z= 1- x2-y2   so, A/c to stroke theoram…

Q: Given F(x, y, z) = xzi + 3yj + yzk be a vector function that passes through a surface S of z+x² + y?…

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Q: Use Stokes' theorem to evaluate curl F. dS, where F(x, y, z) = e*Y cos(z)i + x²zj + xyk, and S is…

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Q: Let (S) be the surface given by the following equation. x²+y? = 1+ z?. An equation of the tangent…

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Q: Let F(x.y.z)=-y²i+xj+z² k, S is the shaded part in the figure at right and C the boundary curve of S…

A: Stroke's theorem: Let S be an oriented smooth surface that is bounded by a simple, closed, smooth…

Q: 3) Use Stokes theorem to evaluate (| curl F-ndS. where F= (xyz, x, e" cos z), while S is a surface…

A: r(t) is the vector function for C, the curve around the base of the hemisphere (and therefore the…

Q: Given F(x, y, z) = xzi + 3yj + yzk be a vector function that passes through a surface S of z+x? + y?…

A:

Q: Let G(x, y, z)= (1+9xy)2 and consider a surface S given by the parametrization r(u, v)= [u, u', v],…

A: According to the company policy, we are supposed to answer the first question only. For the rest…

Q: Use Stokes' Theorem to evaluate [[(curlF)•ndS where F(x,y,z) =x’i+z°j-y°k and o is that portion of…

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Q: Let F = [4z?, x2, 2y²] and C be the positively oriented boundary of the triangle with vertices (0,…

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Q: Let F(x, y, z) = zi + xj – yk and the curve Cbe the boundary of that portion of the surface S above…

A: Given : vector field F→x, y, z=z i^+x j^-y k^ and the curve C be the boundary of that portion of the…

Q: Evaluate the line integral in Stokes' Theorem to find the surface integral [[ (V × F) · ndS, where…

A: Given: The vector is Fx,y,z=4y,-xz,exyz. To find: The value of∬S∇×F·ndS by using the line integral…

Q: 3- Use Stokes theorem to compute the line integral Sc F dr, where F = [e-3y, e², e2"] and C is the…

A: According to the given information, it is required to use the stokes theorem to compute line…

Q: 5. Use Stokes theorem to evaluate J[V x F-ds over the surface of the paraboloid z = 5-x² - y² above…

A: If S  be   smooth surface bounded by a  closed , positively  oriented  smooth boundary curve C with…

Q: Let ~F = (−xy + xz2)~i + (4xz + y2)~j + ez ~k. Use Stokes’ Theorem to calculate ∫C ~F ·d~r, where C…

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Q: Let F= [e, e?, e*]and C be the boundary of the surface S: z x2, 0sxs2,0sys1. If we use the theorem…

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Q: Use Stokes's Theorem to show that 0= [[ curl(F) - dS where F(r, Y, z) = (2, 2ry, x + y) and S is the…

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Q: Given F(x, y, z) = -xị + yzj + xyzk be a vector function that passes through a surface S of z+x² +…

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Q: Let F xyzi + (y + 1)j + z°k, and let S be the surface of the unit cube 0s x, y, z <1. Evaluate the…

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

Q: #6. Use Stokes' Theorem to evaluate S. F dr, where F(x, y,2) = i+(x + yz)j+(xy-V) k and C is the…

A: Given that, Fx,y,z=i+x+yzj+xy-zk and C is the boundary of the part of the plane 3x+2y+z=1 in the…

Q: 5. (a) Evaluate the surface integral / :² dS where o is part of the cone z = V + y that lies between…

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Q: Use Stokes' theorem to calculate , F.ds, where F = (z, x, y) and C is the boundary %3D of a triangle…

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Q: Jse Stokes' Theorem to evaluate F. dr where F(x,y, z) = xi + yj+ 2(x² + y²)k and C is the boundary…

A: answer is in next step

Q: Let a : D → R° be the parametric surface defined by D = {(u, v) E R² | u E [0, 1], v E [0, 1]} and…

A: Solution:-

Q: 3- Use Stokes theorem to compute the line integral ScF · dr, where F = [e-3y, e², e-2"] and C is the…

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Q: Consider a surface S which is parameterised by {r(u, v) = ui + (u + v)j+ vk :0 < u < 1,0 < v< 1}. At…

A: We have a surface S={r→(u,v) =ui^+(u+v)j^+vk^ :0≤u≤1, 0≤v≤1} Normal vector to the surface S is given…

Q: iv) Given F (x, y, z) = -yzį + 2xj + xyzk be a vector function that passes through a surface S of z+…

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Q: 2). Use Stokes' theorem to evaluate the integral 6 F • dř for F(x, y, z) = (x – y)î + (y – z)ĵ + (z…

A: Formula: By Stoke's theorem we have ∮C F · dr¯  =∫t=t1t=t2 F¯(r(t))r'(t) dt, where r(t) is…

Q: Use Stokes' Theorem to find the line integral (x+ 2y°)ï+ (y + z²)j+ (z + 2r²)k) · dř where C is the…

A: To find the line integral, Using stokes theorem.

Q: Use Stokes' Theorem to evaluate |/. CurlF dS, where F(r, y, 2) = xyzi + xj + e" cos zk and S is the…

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Q: Use Stokes' Theorem to evaluate curl F. ds. F(x, y, z) = xyzi + xyj + x²yzk, S consists of the top…

A: The given problem is to evaluate the surface integral using stokes theorem where S is the surface of…

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.