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 beneath the plane : = 3, oriented using a downward pointing normal. Compute the flux of F through S, i.e. the surface integral || F.n dS S (A) 67 (B) – S14 (C)-97 (D) 817 4 (Е) -187 |

Q: 1) Consider the vector field F(x, y,z)= 2xyi+(x² +z² )j+2 yz k . a) Find div(F(0,2,3)) b) Find curl…

A: topic - divergence and curl of a vector a) 8  b) 0

Q: 2.10 (a) Express the vector field H = xy'za, + x'yza, + xyz'a. in cylindrical and spherical…

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Q: In problems 11 and 12, do the computation two ways: i) by using the definition of either flux of a…

A: Using i) the definition of the flux across the surface is  ∬SF.dS.

Q: Find the value of the line integral ∫CF→⋅dr→if F→(x,y)=(y^3+11)i→+(3xy+11)j→is a conservative vector…

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Q: %3D 15. If the vector field F(x, y, z) = xz i+ xyz j-y k, find curl F.

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Q: 1. The value of ls F dS for the vector field F= where S is the surface of the region E that is the…

A: Given vector F⇀=x2z3 i^ +2xyz3 j^ + xz4 k^ By Divergence theorem we know that ∫∫sF.⇀dS⇀=∫∫∫V∇.F ⇀dv…

Q: In Exercises 5-9 compute the flux of the vector field F through the parameterized surface S. -F = yi…

A: We have to find the flux of the vector field.

Q: (b) Compute the flux of the vector field F(x, y, z) = y i– 2.xyk over the triangle T with vertices…

A: Given vector field is: F→=yi^-2xyk^ Given vertices of the triangle are: 1,0,0, 0,1,0 and 0,0,2 Then…

Q: 11. Suppose that E is the solid upper hemisphere of radius a > 0, whose base is a circular re- gion…

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Q: F(1, y, 2) = (2c" cos(3y), –3e²" sin(3y) + VV:). a. Find divF(x, Y, 2) and curlF(x, y, z). b. Based…

A: This is a problem of multi-variable calculus.

Q: 6 The flux of the curl of the vector field F(x, y, z) = (y², r, 2) through the surface Σ = {(x, y,…

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Q: a) Is the vector field F = (a2 - yr, y?- xy) path independent (conservative)? %3D O True O False b)…

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Q: 2 Let CCR² be the circle with radius 6 centered at the origin. Let F : R² → R² be the vector field…

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Q: 5. Find the outward flux of the vector field F(x, y,z) = x²i + y°j+z²k through the first octant…

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Q: 2. Let E be the piece of the cone z = 'outward' (pointing away from the z-axis) normal. Let F be the…

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Q: 34. F = x³i + y*3 + (x² + y²) k and S is the closed cylinder x? + y² = 4 with 0 < z < 5.

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Q: (b) For the case a = 3, b = 4, and for the constant vector field F(x,y,z) = 4 1 + k , find the flux…

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Q: (b) Determine the value of a for which F(2, y, 2) = (2x°y + z²)i+ (xy³ – x°z)j + (axyz – 20°y^)k is…

A: b)Solution: Given that

Q: Problem 2. Let F be the vector field (re² + cos(x), –ye², z sin(x)), and S the paraboloid z =1-1² –…

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Q: In Problem 3, find the flux of the given vector field F through S according to the described…

A: Given data:  Vector field F→=yi→+2xj→+3zk→ S is the portion of the cylinder Radius of cylinder…

Q: VII. - ) Let C be the curve consisting of the line segment from (0,0) to (2,0), the the portion of…

A: Given that a curve consisting a line segment from (0,0) to (2,0) and a portion of circle from (2,0)…

Q: Question 2 The out-flux of the vector field F(r, y, z) = (sin(2x) + ye*, (y + 1)², –2z(y + cos(2x) +…

A: By Divergence Theorem, the flux of F→ through the surface of D is given by ∬SF→·dS→=∭DdivF→ dV.…

Q: Question 3 (4-u (2y+z -5y over surface S: f(u, v) =2+v ), (u, v) E [-1,1] x Cive surface integral of…

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Q: Question 5. Consider the vector field F= 2° + y* + 2*)} (z, v, 2). Using the component test, verify…

A: (a) Choose two points in 3-D plane. A=a, b ,c , B=p , q ,r , C=(1 , 2 ,3) , D=(2 , 5, 8) .If the…

Q: Find the flux of the field F(x, y, z) = (x, y, -2z) through the surface z = ! =1– 2² – y² (0 < ¤< 1,…

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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: Problem EL – A3. Compute the flux of the vector field F(x, y, z) = (x³, y³, 2³) over the surface…

A: Ad per our guidelines we can solve only one question at a time. Please repost unsolved problem…

Q: 1. Consider the function f (x, y, z) = y- + y² +z? Determine the gradient vector field of f, F =Vf…

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Q: Question 3 (2y +z z2 Give line integral of vector field v(x,y,z) = along curve y: ř(t) = | -t),t E…

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Q: 2. – 1)k upward across the rectangular surface 0<x< Find the flux of the vector field F(x,y)= xi +…

A: Consider the given information: F→(x,y)=xi→+yj→+(x2-1)k→ The rectangular region is give as: 0≤x≤2,…

Q: 4) Consider the conservative vector field given by: F (x, y) = (y² – ye“, 2xy – e*) A potential…

A: Given vector field is F→x,y=y2−yex,2xy−ex. We have to find the potential function that generates the…

Q: 1. Let F =i+y^2j +z^4k. Find the general flow lines for this vector field. Present the flow line…

A: Consider the provided question,

Q: Find line integral fF. dr of conservative vector field F (x, y, z) = 'i+ j+-k where C is a path from…

A: Find the line integral of a conservative vector field

Q: Problem 50. Using the Divergence theorem to find the inward flux of the vector field F(x, y, z) =…

A: Given that the vector field,

Q: 2.) Calculate the curve integral over the vector field V(7, y) = ( -x*y xy² for the positive…

A: Green's theorem

Q: Find the Curl of the vector field. F(x,y, 2) = 3e" sin(y)i+ 7e* cos(y)j+ &z k cos (y)i+ &z k. %3D…

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Q: (x - x²y, xy²) along the boundary of D = {(x, y) E R : x² + (y – 1)² < 1}, counterclockwise…

A: Circulation of the vector field

Q: I have trouble getting to the answer. Just wondering if I can get it step by step.

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Q: - L ---- Let F be the vector field (r², 2r, z²). Let C be the boundary of the surface S given by 4x²…

A: Given- Let F→ be the vector field x2, 2x, z2. Let C be the boundary of the surface S given by 4x2 +…

Q: 9. (~Ch16 Review #13) Find the work done by the conservative vector field F(x, y) = along the curve…

A: given a conservative vector field  Fx,y=4x3y2-2xy3,2x4y-3x2y2+4y3 and a curve  rt=t+sinπt,2t+cosπt…

Q: Consider again the vector field F = (x + y)i + (x² + y² )j. What is the downward flux across the…

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Q: 16. Compute the flux V· nds of the vector field V(r, y) = (x² – 2ry, y² – 2xy) across the curve C :…

A: We have to compute the flux ∫ℂV·nds of the vector field Vx,y=x2-2xy,y2-2xy across the curve…

Q: Question 2 The out-flux of the vector field F(r, y, z) = (sin(2.r) + ye, (y + 1)², -2z(y+ cos(2a)…

A: There is no such introduction for the solution.I have explained it within the solution.Please go…

Q: 2. Find the circulation and flux of the field F(r.y) = (-y,r) along and across the curve 7(1) =…

A: We have to find the circulation and flux of the field F(x,y)= (-y,r) along and across the curve…

Q: 6) Given a hemisphere bounded by S,: z = /25 – x2 – y? and S2:z = 0.Find the outward flux of the…

A: The flux of a vector-valued function F(x,y,z)=F1i+F2j+F3k over a surface S is defined as ∫SFdS.…

Q: 12. Find the flux of the field F(x, y, z) = 4xi + 4yj + 2k outward (away from z-axis) through the…

A: Flux of the given field is given by

Q: EXAMPLE 4 Line integral of a conservative vector field Evaluate |(2ry – z) i + (x² + 2z) j + (2y –…

A: The line integral of a conservative vector field is  C is a simple curve from A(-3,-2,-1) to…

Q: Consider the vector field F (x+ y)i + (x² + y²)j What is the downward flux across the line segment…

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Q: 1a) Given a conservative vector field F = (A, B , C) such that A = 2zx, B = 3z + 2y, C(0,0,0) = 1…

A: We have to solve three sub-parts :        (a) Given a conservative vector field F = A , B , C such…

Q: 3. What is true about the value of F.dS for the vector field F (r'y,-ry,-I*yz) where S is the…

A: Solution 3: Given  F=x3y,-x2y2,-x2yz We have to calculate ∬SF.dS, 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: