Use the Stokes' Theorem to caleulate the circulation of the field F = -6y*7 + 6x*J° + 7z*R° around the curve C: the portion of the paraboloid x² +y² = z cut by the cylinder x² + y² = 4. %3D

Q: F= (y, 2x, -z) and C is the circle x² +y² = 9

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Q: Use Gauss' Divergent Theorem to calculate F. dS. where F = (sin(y?) + a)i + ey?j – 2e*+cos² (au)k…

A: Gauss divergence theorem is used

Q: Let S be the part of paraboloid z = 9 – x2 - y? with z 2 0. Verify Stokes' theorem for vector field…

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Q: Use Stokes' Theorem to evaluate curl F. ds. F(x, y, z) = xyzi + sin(xyz)) + xyzk, S is the part of…

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Q: Use the surface integral in Stokes' Theorem to calculate the circulation of the field F = (y +z*)i+…

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Q: Compute the line integral of the vector field F(r, y, z) = (x + 2y + 3z) i+ (2x + 5y)j+(z – 3y)k…

A: As per policy, we will solve the Question 4 only please repost it to get solution of Question 5. In…

Q: Use Stokes' Theorem to evaluate curl F· dS. F(x, y, z) = 4y cos(z) i + e\ sin(z) j + xeY k, S is the…

A: Given that Fx,y,z=4ycoszi+exsinzj+xeyk and S is the hemisphere x2+y2+z2=9,z≥0. The expression for…

Q: Use Green's Theorem to find the work done by the force field F(r, y) = (e" – y³)î+ (x³ + cos y)ĵ on…

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Q: Use Stokes' theorem to evaluate curl(F) · dS. F(x, y, z) = exy cos(z)i + x2zj + xy k, S is the…

A: F(x, y, z) = exy cos (z) i + x2 z j + xy k

Q: Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated…

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Q: Use the surface integral in Stokes' Theorem to calculate the circulation of the field F = +z)i+ (x…

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Q: use the surface integral in Stokes’ Theorem to calculate the circulation of the field F around the…

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Q: use the surface integral in Stokes’ Theorem tocalculate the flux of the curl of the field F across…

A: Given  F=(y-z)i+(z-x)j+(x+z)k S:r(r,θ)=(rcosθ)i+(rsinθ)j+(9-r2)k 0≤r≤3, o≤θ≤2π

Q: Let a, b > 0. Verify Stokes' Theorem f F - dĩ = SS, curl(F) - ndS for 2xz the vector field F(x, Y,…

A: Let a, b>0. Verify Stokes' Theorem ∮F→· dr→ =∫∫σ curl F→. n dS for the vector field F→x, y,…

Q: Use Stokes' Theorem to calculate the work done by the field F through the shell S. F = x 4i + 7x j…

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Q: Use the surface integral in Stokes' Theorem to calculate the circulation of the field F=xi+ 4xj +z*k…

A: Find the explanation below

Q: 3. Use the surface integral is Stokes' Theorem to calculate the flux of the curl of the field F…

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Q: Consider the vector of electric field E(x, y, z) = (sinh x)i – (cosh y)j – xyz k a. Compute the…

A: The given vector field is, E(x,y,z) = (sinh x) i - (cosh y) j - (xyz) k   the divergence and curl…

Q: ) Verify Stokes' Theorem for the hemisphere S : x²+y²+z² = 9, z > 0, %3D its bounding circle C: 1?…

A: Stokes theorem  If a smooth surface S bounded by closed ,smooth boundary curve C oriented in…

Q: Use Stokes' Theorem to evaluate || curl F. dS. F(x, y, z) = x² sin(z)i + y²j + xyk, S is the part of…

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

Q: Use Stokes' Theorem to evaluate curl F· dS. F(x, y, z) = zeYi + x cos(y)j + xz sin(y)k, S is the…

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Q: Find the surface integral of the field F(x,y,z)=9xi +9yj +9zk across the portion of the sphere x² +…

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Q: Evaluate the surface integral /F- ds for the veetor field F = (y. -r, 2:), where S is the hemisphere…

A: Surface integration

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: Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated…

A: Let F be a vector field and C be the boundary of the surface S. The Stoke's theorem states that…

Q: Use Stokes' Theorem to evaluate ∫∫ s  curl F. dS  F(x,y,z) = x2 sin zi +y2 j +xyK, S is the part of…

A: Given: F(x,y,z)=x2 sin z i + y2 j +xy k and S is the surface of paraboloid z=1-x2-y2, which is above…

Q: Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field F across…

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Q: Let S be the surface in the xy-plane defined by the inequalities |æ|+ |y| < 1, k. Verify the Stokes…

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Q: Let ƒ(x, y, z) = (x2 + y2 + z2)^(-1/2). Show that the clockwise circulation of the field F = ∇ƒ…

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Q: Evaluate the flux integral ff. curl (z cos x i+yj+x sin y k) ds where S is the part of the outward…

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Q: Let us verify Stokes' theorem using the vector field F = (x 2 - y)i + 4zj + x 2k, where the closed…

A: We are given a 3d vector field and a cloced conture, using which we have to justify the Stoke's…

Q: Evaluate the surface integral ff, F · dS for the given vector field F and the oriented surface S. In…

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Q: Use Stokes' theorem to evaluate curl(F) · dS. F(x, y, z) = x² sin(z) i + yj + xyk, S is the part of…

A: Given that, And he paraboliod is The aim is to find The radius is 3. Take C is x(t)=cos t and…

Q: Verify that Stokes' Theorem is true for the given vector field F and surface S. F(x, y, z) = -yi +…

A: Concepts introduction: Let S be an oriented smooth surface that is bounded by a simple, closed,…

Q: Use the Divergence Theorem to compute the net outward flux of the field F = (x,y,z) across the…

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Q: Use Stokes' Theorem to evaluate SSs curl F· dS. F(x,y,z) = x2z2 i + y²z² j + xyz k S is the part of…

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Q: Let S be the surface in R' that is the part of the ellipsoid 2x2 + y² + z? = 2 satisfying y > 1.…

A: In this question, we use the div.  theorem to evaluate the surface integral. given that…

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: Consider the closed, solid region E between the paraboloid z = 4 – x? – y? and the xy-plane, and…

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Q: Use Stokes' Theorem to evaluate curl F· dS. F(x, y, z) = x² sin(z)i + y²j + xyk, S is the part of…

A: Given problem is :

Q: Given the radial field (x, y, z) r (x2 + y? + z2)3/2 r|3/2 Find the outward flux across the sphere…

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Q: (4) Use Stokes' Theorem to calculate the work-done by the force field F(r, y, z) = (x² + z²)î + (y²…

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Q: Evaluate the flux integral , curl (z cos x î + y j+ x sin y k) dS where S is the part of the outward…

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Q: Use the surface integral in Stokes' Theorem to calculate the circulation of the field F = by the…

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Q: Use Stokes' Theorem to evaluate curl F. dS. F(x, y, z) = exY cos(z)i + x²zj + xyk, S is the…

A: Given F→x,y,z=exycoszi^+x2zj^+xyk^ We have t find the integral ∬ScurlF→⋅dS, where S is the…

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: Use Stokes' Theorem to evaluate|| curl F. dS. F(x, y, z) = x²y³zi + sin(xyz)j + xyzk, S is the part…

A: The vector field is Fx, y, z=x2y3zi+sinxyzj+xyzk Find: ∫∫Scurl F·dS Then, the double integral…

Q: Verify Stokes Theorem for the vector field F(x,y)=(-2yz) i + y j+3x k on the part of the paraboloid…

A: Given F→x,y =-2yzi→+yj→+3x k→  We have to Verify the Stokes theorem for the given vector field on…

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…

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: