Sketch the surface S and evaluate F N dS for the vector field F(x, y,z) = x³i + y°j+ 2z k where S is the surface of the region bounded by z = 4 - /x² + y2 and the xy -plane.

Q: Consider the vector field Fˆ(x, y, z) = (y/(x^2 +y^2) i - (x/(x^2 +y^2)j (a) compute the curl (b)…

A: Hello. Since your question has multiple parts, we will solve the first three parts for you. If you…

Q: State the theorem for Vector Surface Integral.

A: More precisely, the divergence theorem states that the surface integral of a vector field over a…

Q: Consider the function f(x, y) = (e - 3x) cos(y). Suppose S is the surface z = f(x, y). (a) Find a…

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Q: Your friends correctly calculate the gradient vector for f(x, y)=x²+y at the point (2,-3) as…

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Q: Find div(curl F) = V. (V x F). F(x, y, z)= xyzi + yj + zk

A: If vector field then from the question

Q: The outward flux of the vector field F(x, y, z) = (x, y, z') across the boundary of D- {(x, y. z)…

A: Using divergence theorem calculate the flux:  φ=∯SF.dS   =∰V∇.F.dv…

Q: Compute the flux of the integral of the vector field F(z,y, 2) = (2,y, 2) through the half cylinder…

A: Given vector field is, Fx,y,z=x,y,z. It can be written as,  Fx,y,z=xi^+yj^+zk^. Flux of vector field…

Q: II. Let S be the portion of the cylinder x² - 6x + z = 4, oriented by upward-slanting normal…

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Q: Find a vector valued function which gives the curve of intersection of  y2 + z2 =16  with the plane…

A: Given the curves:

Q: 4. Consider the vector function r(z, y) = (x, y, x² + 2y*). (a) Re-write this vector function as…

A: For the given vector function, PART A The surface function is given by, f(x,y)=xi^+yj^+(x2+2y2)k^…

Q: 'I' F. ds for the given vector field F and the oriented surface S. In other words, find the flux of…

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Q: Find the flux of the vector field F(x, y, z) = across the surface σ, where σ is the surface of the…

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Q: Solve using method of undetermined coefficients: y" + y' - 2y = x². %3D

A: Solution :-

Q: The gradient vector for the surface xy + xz + yz = 1 at the point (1, 1,0) is (1, 1,0). Select one:…

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Q: Compute the flux of the vector field F(x, y, z) = (2,0, x³) across the portion of the plane x-z = 0,…

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Q: A vector field is given as F=3x²i – yj + z'k a) Find the streamline of the vector field and also the…

A: Streamline of the vector field is the family of curves which are tangent to the vector field. Curl…

Q: 5 The flux of the curl of the vector field F (1. y, z) = (y², 7, z) through the surface E= {(z, y,…

A: To find- The flux of the curl of the vector field Fx, y, z = y2, x, z through the surface ∑ = x, y,…

Q: Find a unit normal vector to the surface 6x2 + 2y2 + z2 = 225 at the point P : (5, 5, 5), and…

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Q: Use Divergence Theorem to evaluate the flux integral ∫∫F·ndS, where F = 〈x^2, y^2, z^2〉, S is the…

A: Divergence theorem:∫∫sF.⇀  N^dS =∬∫DdivF⇀dV , here D is the solid bounded by the closed surface S.

Q: Find the total flux of the vector field F= (x^2 z)i + (y^2 z)j+(z^3)k through the closed surface of…

A: Given vector field is  F=x2zi+y2zj+z3k              (i) Given surface is  x2+y2=16z=2z=-2 Let us…

Q: Find the flux of the vector field F(x, y, z) =(x)i+(y)j+(z)k through the surface S of the…

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Q: Compute the flux of the vector fields F(x,y, z) = (, y, z) across the 1 and z = 4 whose vertical…

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Q: 7) Consider the vector field F(x, y,z)= (y,z,x). Find the flux of this vector field across the…

A: Flux = double integral of (F(r(u,v)).ru*rv) dvdu

Q: F (x,y,z) =(2x+ cos(yz) )i +(xe^-z -y)j+ zk vector field D: x^2 + y^2 =4  ,  x+z=2  , z=0…

A: Given vector field is, Fx,y,z=2x+cosyzi+(xe-z-y)j+zk.

Q: Find the gradient vector field of flx, y, z)=x'y+y%z4+xz6 O V flx, y,z)= (3x?y+z°, 5x³y³ + 4y*zA,…

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Q: Surfaces Parameter irina parater

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Q: Verify the Divergence Theorem for F = (-2, 3, 2³) across the surface of a sphere of radius 4 with…

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Q: Let S be the portion of the surface x + y“ + 2z“ = where y > x. Suppose S is oriented so that the…

A: Given , the surface x2+y2+2z2=1, where y≥x, and the unit vector n⇀=0,1,0  and the ∇×F=-2y2,2z2,sinx

Q: Use Green's theorem for the work done by the force F= (x - y)i + (x + y)j, for the shaded region R…

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Q: This surface (dreamed up by Steiner while in Rome) looks smooth but is non-orientable, with the…

A: Given: The parametrization for the surface is given by σ(u,v)=x,y,z=sin2usinv,sin2ucosv,cos2usin2v…

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: