Question 31 Find the flux of F through S, F NdS where N is the upward unit normal vector to S. F(x, y, z) = xi+yj+zk, S: z = 36–x² - y², z 20 972 T O 1,944 T O 2,527 TT O 1,361 7 O 583 77

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Q: Find the flux of F across S, F.N ds where N is the upward unit normal vector to S. F(x, y, z) = xi +…

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Q: Consider the vector field Ĥ(x, y, z) = (y², x, z²). Let S3 be the portion of the paraboloid 2 = x² +…

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Q: (b) Let D be the disk of radius a in R², and let C denote it's boundary. Let N be the unit normal…

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Q: QUESTION 14 Let S be the upper half of the unit sphere X+y+z%31 with outward-pointing normal vector…

A: Sphere  x2+y2+z2=1

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Q: Find the flux of F across S,       SF · N dS where N is the upward unit normal vector to S. F(x, y,…

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Q: F.dr. QUESTION 12 Let C be the oriented curve parameterized by r(t)=(t,t> for 0sts2 (so C is…

A: Vector calculus

Q: Question 5. Let F be a force field in space given by F(x, y, z) = (yz + y² + z² , 2yx + xz , xy+…

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Q: Question 7 a) Consider the vector field F(r, y, z) = (2ry, r² + 2z, z°y²). %3D Find curlF.

A: Since you have asked multiple questions in single request so we will be answering only first…

Q: Question 5 Let F: R2 -R? be the vector field defined by F(r, y) = (3x cos y +2 cos r, -r sin y). %3D…

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Q: 1. Find the unit normal vector to the surface z = Vx2 + y? at the point (3, 4, 5).

A: A detailed solution is given below

Q: Question No. oy: Evaluate I = SFinds where f= zi+ j+ 34 3k and n is unit vector. S is the sureface…

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Q: Question 2 Find the flux of the vector field F across the surface S in the indicated direction. F =…

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Q: Find the flux of F across S, F.N dS where N is the upward unit normal vector to S. F(x, y, z) = xi +…

A: The given integral is ∬SF·N ds , where S: z=a2-x2-y2  and F(x,y,z)=x i^ + y j^- 2z k^ .....1. To…

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Q: Question A7 Calculate the flux of the vector field F = (0, z, y) through the surface E: x = 2u?v? E:…

A: Value of the flux depends on the normal direction. So it may be positive or negative .

Q: (b) Let D be the disk of radius a in R?, and let C denote it's boundary. Let N be the unit normal…

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Q: 19. Suppose that the normal unit vector N of a surface S at z = f(x, y) is determined, how does the…

A: Note: As company policy, we solve first question only. Please resubmit the rest of the questions in…

Q: QUESTION 5 Let S be the upper half of the unit sphere X+y²+z =1 with outward-pointing normal vector…

A: please see the next step for solution

Q: Find the flux of F across S, F.N dS where N is the upward unit normal vector to S. F(x, y, z) = xi +…

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Q: Find the principal unit normal vector N for the curve r(t). r(t) = (3 + t)i + (7 + In(cos t))k, -1/2…

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Q: Problem #9: Let F be the vector field F(x, y, =) = xi+ yj+ 2=k and let S be the part of the cone - 2…

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Q: Find the flux of F acros S, F N dS IS where N is the upward unit normal vector to S. F(x, y, z) = xi…

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Q: 10. Verify Stokes' Theorem, fF.dr=ffcurl(F) ds, for the vector field F= (z, 2x, 2y) and the surface…

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Q: Find the flux of F across S, F N dS where N is the upward unit normal vector to S. F(x, y, z) = 10i…

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A: Please check step 2 and 3 for the solution.

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Q: Question 6 A vector field is given by F(r) = zi+a+yk, and a paraboloid is given by S = {(x, y, z) €…

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A: Stokes theorem is used to evaluate integration

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Q: Question 10 Use Stokes' theorem to verify that where the vector fields (V x F). ds S oriented…

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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: