Verify Stokes' theorem for F = (x+y-4) i+3xy j+ )k, over the surface of the hemisphere x2 + y2 + z2 = 16 xy-plane?

Q: Use stokes theorem to evaluate f.dr where f = (4z+y, 3x, 7y) and C is the curve of intersection of…

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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's Theorem by evaluating F.dr as a line integral and as a double integral. F(x, y, z)…

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Q: Use Stokes' Theorem to evaluate F• dr where C is oriented counterclockwise as viewed from above.…

A: Given:     Stokes'Theorem :                            ∫cF·dr =∬scurlF·ds   If F=Pi+Qj+Rk,…

Q: Use Stokes's Theorem to evaluate f.7 X F.dŠ, where F (x, y, z) = (y +29)î – xj + x³yk and S is the…

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Q: Let F = (-z, x, y²). Find curi Fds, where S is the paraboloid z = x2 + y?, 0 < z< 4. (You can do the…

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Q: Find the tangent plane of the surface z = f(x, y) at (0,0, 0) in which 3y f(2, y) = || cos /3 t dt.

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Q: Verify Stokes theorem for F = (xy -x2)î + yx?j in the region in the xy-plane bounded by y = 0,x = 1…

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Q: Use Stokes' Theorem to caleulate ſ. F - dř, where F = zyi+3=3+5zk, and C is the closed path around a…

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Q: ix of F across the surface defined by R(u, v) = (2u, 3v E [0, 1].

A: We have to find the flux according to the given information

Q: Use Stokes' Theorem to evaluate [F•dF.C is the boundary of the portion of :=5-x² - y above the…

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Q: This under subject of Linear Algebra and Vector Analysis

A: Given : z=1-x2-y2             F ⇀ =z-yi^+x -zj^  +x-yk^ Verify : Stokes's Theorem

Q: Use Stokes' Theorem to evaluate curl F· dS. F(x, y, z) = tan-(x²yz²)i + x²yj + x²z?k, S is the cone…

A: The objective here is to evaluate: ∬ScurlF⋅dSWhere, Fx,y,z=tan−1x2yz2i+x2yj+x2z2k; and S is the cone…

Q: ns: Find the tangent plane to the surface of x2 + z² = y² + 9 at (1, –1,3). 3

A: Follow the procedure given below.

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: Use Stokes' Theorem to compute 1 z? dx + (ry) dy + 2020 dz, where C is the triangle with vertices at…

A: Given I=∮C12x2+xydy+2020dz where C is the triangle with vertices at 1,0,0 , 0,2,0 , 0,0,2 traversed…

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: Let S be the cylinder x2 + y2 = a2, 0&lt;=z &lt;=h, together with its top, x2 + y2 &lt;=a2, z = h.…

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Q: Compute (VxF). dS, where S is given by x² + y² + (z − 1)² = 2 and z ≥ 0, F(x, y, z) = (-esinzy, x +…

A: The general form of a vector-valued function is given by F1,F2,F3 where F1,F2 and F3 are the scalar…

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: 63. Verify Stokes' theorem for A = (y - z +2)i + (yz +4)j – xz k, where S is the surface of the cube…

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Q: Let F < xy, 6z, 3y Use Stokes' Theorem to evaluate F · dī, where C is the curve of intersection of…

A: The given problem is to evaluate the line integral using stokes theorem, we have to find the limits…

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: 18 Verify Stokes' theorem F.dr = (curl F)· dS C S where F = (x² + y – 4, 3xy, 2xz +z?) and S is the…

A: We have

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: Use Stokes' Theorem to evaluate F• dr where C is oriented counterclockwise as viewed from above.…

A: Here use the stokes theorem to find the integral of the function.  And given that the function…

Q: (ii) Use Stokes' Theorem to evaluate SF · dr, where F(x, y, z) = x²zi+xy²j+z²k and C is the curve of…

A: Given function is F(x,y,z)=x2z i^+xy2 j^ +z2k^ . Plane: x+y+z=1 Cylinder: x2+y2=9 The objective is…

Q: 1. Compute ff,(V × F) - dS, where S is given by r² + y? + (z – 1)? = 2 and z > 0, and F(x, y, z) =…

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Q: Use Stokes' Theorem to evaluate the line integral (y + 2z)dr + (x + 2z)dy + (x+ 2y)dz where C is the…

A: Given intgeral. ∫Cy+2zdx+x+2zdy+x+2ydz The Curve is formed by the intersection of the sphere…

Q: Use Stokes' Theorem to evaluate |/ curl F. dS. F(x, y, z) = tan-(x²yz²)i + x²yj + x²z?k, S is the…

A: Stokes' Theorem:                                 ∮C F .dr =∬S curl F. dS Where C is the boundary of…

Q: Use Stokes' Theorem to evaluate L* + cos x) dx + (sin y + z²) dy + x dz where C is the closed curve…

A: To Evaluate: Use Stoke's Theorem:

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: Use Stokes’ Theorem to evaluate the line integral  ∮  F·d r where F = 〈x^2−z^2, y, 2xz〉, and C is…

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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: 3. Suppose F z = 4 – a? – y? above the r- y plane, oriented upward. Calculate fe F dĩ in two ways:…

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Q: Verify stokes theorem for F = (2x- y)i – yz'j - y zk over %3D the upper half surface of the sphere…

A: Given the field, F¯=2x-yi¯-yz2j¯-y2zk¯ S is the upper half surface of the sphere x2+y2+z2=1 bounded…

Q: '. Use Stokes' Theorem to compute || CurlF •dS for the hemisphere x + y +z² =9, z20, intersected…

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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 ~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: 11. Consider F(x, y, z) = (-y, x, e²) and the surface defined by S = {(x, y, z) : z =1– 2² – y², x²…

A: Given F=-y,x,ez and S=x,y,z| z=1-x2-y2,x2+y2≤1. Verify the stokes theorem ∫CF·dr→=∬Scurl F·n→ dS.

Q: 2. When F = (y z, xz, x² y?) and S is the part of the paraboloid z = x? + y? that lies in the…

A: First, we have to parametrize the surface to apply the Stokes theorem.  Part of the paraboloid is…

Q: 1. Use Stokes' Theorem to evaluate curlF.dS where F =z'i+ (4y-y) j+2zk and S is the portion of z = r…

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Q: Find the plane tangent to the given surfaces at the specific point. f(x, y) = ex(x cos y − y sen y)…

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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…

Q: Define stokes theorem and evaluate the line integral SF.r'(s) ds where F = [4z, 2x, 2x], C: the…

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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 compute | 1 F. dr where C is the curve where the cylinder a? + y? = 1…

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

Q: Verify Stokes's Theorem by evaluating S F(x, y, z) = (-y + z)i + (x − z)j + (x − y)k S: z = √1-x² -…

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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: