Verify Stokes' Theorem for F = xyî + yzj + xzk and where s is the surface of the paraboloid z = x² + y² for x² + y² < 9.

Q: Evaluate the surface integral F. ds where F = 5xyi +8x²j+ 5yzk and S is the surface z = ïe², 0 ≤ x ≤…

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Q: 4. Use Stokes’ theorem to evaluate fa F. dr, where F(x,y, z) = (xy,yz, zx), and C is the triangle…

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: Verify Stokes' theorem for F = (x+y-4) i+3xy j+ )k, over the surface of the hemisphere x2 + y2 + z2…

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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: Verify Stokes's Theorem for F = z²î+ x²j + y²k and S is the surface z2 = x2 + y², y 2 0, and 0 <…

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Q: Evaluate the surface integral. z2 S is the part of the paraboloid x = y2 + z² given by 0 sx 5 4 1 V5…

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Q: Find a parametrization of the paraboloid given by f (x, y) = x 2 + y2.

A: To find a parameterization of the paraboloid given by: fx,y=x2+y2

Q: Let (S) be the surface given by the following equation. x²+y² = 1+ z?. An equation of the tangent…

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Q: Find a parametric representation for the part of the cylinder y2 + z² = 36 that lies between the…

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Q: Use Stokes' theorem to evaluate frotF. ds. F(x, y, z) = xyi + e²j + xy²k; S consists of the four…

A: To find- Use Stokes' theorem to evaluate ∫SrotF·dS, Fx, y, z = xyi + e2j + xy2k; S consists of the…

Q: 2. Evaluate f, z³ /x² + y² + z² dV where E is the solid hemisphere that lies above the xy-plane and…

A: Solution:- equation of sphere having centre origin (0,0) and radius=1 x2+y2+z2=1…

Q: Calculate F- dS where F = (3zy', ze", z) S is the surface of the solid bounded by the cylinder y? +…

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Q: Evaluate the surface integral. z2 dS, S is the part of the paraboloid  x = y2 + z2  given by  0 ≤…

A: given the surface integral  z2 dS  where S is the part of the paraboloid x= y2+z2   given by 0≤x≤1…

Q: Evaluate the surface integral. J[x²³x² x²yz ds, S is the part of the plane z = 1 + 2x + 3y that lies…

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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: 2)Use Stokes' Theorem to evaluate [F.dr where E2xyi, y² j‡ 2xzk and Sis the first octant с surface…

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Q: b) Use Gauss theorem to evaluate F.ndS, where F(r, y, 2) = (2x²y² + In z) i – xy³ j – tanh(a² + y³)…

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Q: Use Stokes's theorem to evaluate o F· dr, where F(x, y, 2) = xy² i+x²yj+yz k, where C is a…

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Q: (b) Verify Stokes' theorem on a region which is a portion of the sphere r2 +y² +2 16 which lies…

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Q: Evaluate the surface integral. y ds S is the surface z = 3/2 0 <x < 4, 0 < y< 3 +

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Q: Attached picture is the question

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Q: Let o be the portion of the paraboloid F(u, v) = u cos(v)i+u sin(v)j+ u*k for which 0 < u< sin(v), 0…

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Q: Find the area of the part of the paraboloid z=x² + y² that lies under the plane z=9.

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Q: Use Stokes' Theorem to evaluate curl F· dS. F(x, y, z) = x2z2i + y2z²j + xyzk, S is the part of the…

A: Given: F(x,y,z)=x2z2i+y2z2j+xyzk   and S is the part of the paraboloid z=x2+y2 that lies inside the…

Q: Find the centroid of the surface cut from the cylinder y2 + z2 = 9, z&gt;= 0, by the planes x = 0…

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Q: Let F = < 4x + y², 5y + z, 9z + x So F. dī, Use Stokes' Theorem to evaluate C where C is the…

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Q: 4) Use Stokes theorem to evaluate SF.dr, where с F(x, y, z) = <.xy, yz, *x), where C is the triangle…

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Q: 14. Under the paraboloid z=x² +y2 and above the disk x +y² <9 for (x,y) in the first quadrant.

A: we have to find the volume

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: Evaluate the surface integral. ds 2 + y32), o s x s S is the surface z = x3/2 0sx < 1, 0 s ys2

A: Given,                        z = 23(x32+y32)and domain D : 0 ≤ x ≤ 1, 0 ≤ y ≤ 2.∂z∂x= 23(32 x) =…

Q: - Prove Stokes' Theorem by verifying it for F= [y,z,x] and S the paraboloid. %3D te CEY2=(x, y) = 1…

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Q: Use Stokes' theorem to evaluate f curlF · ds. F(x, y, z) = y²zi+xzj + x²y²k S is the part of the…

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Q: É = ( 2x3 + y°, y3 + z³, 3y²z), S is the surface of the solid bounded by the paraboloid z = 1– x2 –…

A: We will find out the required flux.

Q: Let S be the surface x2 + y² = 4, 0 < z < x + 2. Compute x dx + y dy + dz. S

A: Gauss-Divergence Theorem- Let C be a solid region and S be the boundary surface of C, given with…

Q: Use Stokes' theorem to evaluate V ×F· î dS where = (9y²z, 4xz, 2x²y²) and S is the paraboloid z = x²…

A: Step:-1 Stoke's Theorem:- ∬s∇×F→.n dS=∫F.dr Now, given that F→=9y2z, 4xz, 2x2y2 and paraboloid…

Q: Use Green's Theorem to calculate the work done by the force F on a particle that is moving…

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

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Q: Verify Stokes' Theorum for F(x,y,z) = (y,-x,yz) if S is the paraboloid z = x2 + y2 with x2 + y2 = 1,…

A: Given  Fx,y,z=y,-x,yz Then, F→=yi^-xj^+yzk^

Q: please help me as soon as possbile

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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: 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: The surface area of the part of the paraboloid z=x+y that lies under the plane z=2 is: Z=,

A: Surface area is given as A=∬D1+δzδx2+δzδy2dA

Q: Evaluate the surface integral. ds Is 3/2 + y3/2), o s xs 1, 0sys 3 0sx < 1, 0 s ys 3 S is the…

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Q: se Stokes's Theorem to calculate (curl F) · n dS . 14) F = -2x2yi + 2xy²j + z°k; S is the portion of…

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Q: verify Stokes § = (a%-y)i - yz'i over the uppea hazf SOD Face Of the sphere =1 bounded by the

A: Given that,  F→=(2x-y)i→-yz2j→-y2zk→ To verify Stoke's theorem: ∫CF→·dr→=∬Scurl F→·n^dS Here C is…

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: Use Stokes' Theorem to evaluate , (x + y )dx + (2x – 3)dy + (y + z)dz over the boundary of the…

A: First we will learn about Stokes theorem little bit. If "F1 i + F2 j + F3 k" is a differentiable…

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: