Integrate the vector field h (x, y, =)=ry i+yz j+x k over r (1u) = ui+ j+k, -ןue-L1

Q: Compute the upward flux of the vector field Ĥ(x, y, z) = (x, 0, 2z) across the portion of the…

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Q: Determine the divergence of vector field E = 3x²x + 2zŷ + x²z2 at (2, –4, 0).

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Q: 4. Is this vector field F(x,y)=(ycos x)ĩ +(sin x+ y)j being the path-independent field? Please show.

A: To check if the given vector field is path-independent field or not. Given:

Q: Compute the work done by the vector field F(x, y, z) = (,2, 2yz) when moving a particle: 1+? > a)…

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Q: When is the vector field continuous?

A: given,       Continuous vector field .

Q: Find a potential for the conservative vector field: F(x, y, z) = (2xy + 2x, x² – 2yz³, –6z – 32²y²)…

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Q: What does it mean if the curl of a vector field is zero throughout a region?

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Q: Find the work done by vector field F(x, y, z) = xi + yj + (x + z)k on a particle moving along a line…

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Q: Find the work done by vector field F(x, y, z) = xi + 9xyj - (x + z)k on a particle moving along a…

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Q: Compute the work done by the vector field F = in moving a particle along the circle x + y = 1 from…

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Q: Compute the flux, || Foñds, across the slanted face of the tetrahedron 2 = 4-1- y bounded in the…

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Q: .) Compute the flux of the vector field F = 4x + 4yj through the surface S, which is the part of the…

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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: Why does a two-dimensional vector field with zero divergence ona region have zero outward flux…

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Q: A smooth vector field F has div F(2, 5, 5) = 2. Estimate the flux of F out of a small sphere of…

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Q: Determine the line integral if the vector field is F(x, y) = x²yî – 2xyf On a rectilinear path that…

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Q: Show that the vector field F = 2xyi + (x - 2 y)j is conservative and evaluate the line integral %3D…

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Q: Compute the flux of the vector field (x, - xy*), out of the rectangle with vertices (0,0), (5,0),…

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Q: Find the flux of the vector field F = (x, z, - y) through the helicoid with parameterization r(u, v)…

A: Given :-  The flux of the vector field  F→ =  x3 , z , -y through the helicoid with the…

Q: Determine if the vector field F=(yz,xz,xy) is conservative?

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Q: Find the work done by the force field F(x, y, z) = (x - y2, y – z2, z - x2) on a particle that moves…

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Q: Find the flux through through the boundary of the rectangle 0 < x < 6,0 < y < 5 for fluid flowing…

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Q: Verify the Divergence Theorem in the calculation of the flux of the vector field: through the…

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Q: Compute the flux of the vector field (x2, - xy*), out of the rectangle with vertices (0,0), (4,0),…

A: Given,

Q: Determine whether the line integral of each vector field (in blue) along the oriented path (in red)…

A: it is known that  line integral F·dr will be positive if both F and dr have same direction. and if…

Q: Find a potential functioon for the conservative vector field given by y3/2 3 xy + z, 2 sin z 2.x…

A: According to question,Given vector field,F→x, y, z = 2xx2 + cos z-y322x, -32xy+z, -sin zx2 + cos z+y

Q: Calculate the integral line of vector field F(x,y) around the circle x²+y²=1 oriented…

A: Given that Fx,y=-2yx2+y24+2x,2xx2+y2/4+xy+1. The the integral line of vector field F(x,y) around the…

Q: Why does a two-dimensional vector field with zero curl on aregion have zero circulation on a closed…

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Q: 6. What is the corresponding vector fieldV for the scalar potential p(x, y)= In(x+ ¥)?

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Q: Let divF(x, y, z) = 4x³ + 4y3 +4z³, where F represents the flow of a vector field. Estimate the flux…

A: Given divF→x,y,z=4x3+4y3+4z3radius r=0.1 Centered at point 1,1,1

Q: Suppose a particle travels clockwise on a triangular path connecting the points (0,0), (3,1), and…

A: greens theorem

Q: Compute the flux of the vector field F = xi + yj + zk through the surface S, which is a closed…

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Q: Show that the vector field =(-2 x²y xy²¹ is conservative by finding all its potential functions.…

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Q: Compute the work done by the vector field F(x, y) = (ry, x² + y³) when moving a particle…

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Q: Why does a conservative vector field produce zero circulationaround a closed curve?

A: Let F be an arbitrary conservative field vector around a closed curve C. To prove: Let A and B be…

Q: Calculate the line integral of the vector field F = -2i+ 4jalong the line from the point (3,0) to…

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Q: A smooth vector field F has div F(5, 2, 5) = 2. Estimate the flux of F out of a small sphere of…

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Q: Let W be the region between the sphere of radius 4 and the cube of side 1, both centered at the…

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Q: Consider the vector field F = ⟨ 1 , y ⟩ .Compute the path integral of this field along the path:…

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Q: Compute the flux of the vector field (x3, – xy³), - out of the rectangle with vertices (0,0), (3,0),…

A: Given: F=x3 ,-xy5with rectangle with vertices (0,0) , (3,0) , (3 ,2) , (0,2) To find flux of vector…

Q: The outward flux of the vector field F=(x,0, z) across the cylinder x + y = 1, for 0<z<1, is %3D…

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Q: 1. Is the vector field V (x, y, z) = (2ry, x² – 2², –2yz) conservative?

A: Since you have asked multiple question,as per our guidelines we are supposed to answer only one…

Q: A smooth vector field F has div F(1,4, 4) = 7. Estimate the flux of F out of a small sphere of…

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Q: Let divF(x, y, z) 4x3 + 4y3 + 4z³ , where F represents the flow of a vector field. Estimate the flux…

A: Given that a small sphere of radius 0.1, centered at the point  (1, 1, 1).And                  div F…

Q: If the vector field A = âx3x² + âyy – â̟5z³ is given, express à in cylindrical and spherical…

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Q: Suppose a vector field F(r, y) = construct a potential function o(r, y). i+ j is conservative. Then

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Q: Compute the outward flux of the vector field F = (5xy, 5y²) for the region enclosed by the curves y…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

Q: Calculate F=, around the triangle with vertices: (0,0,2), (2,0,0) and (0,2,0). Can you use Stoke's…

A: Given the vector field F=z-2x,y+2z,-2y+x around the triangle with vertices 0,0,2,2,0,0 and 0,2,0.…

Q: Suppose a vector field F(x, y) = wai + VEj is conservative. Then construct a potential function ø(x,…

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Q: Q3 Show that the vector field F= 2xyi +(x -2y)j is conservative and evaluate the line integral |F•ds…

A: Solution

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: