Find the flux through through the boundary of the rectangle 0 < x < 6,0 < y < 3| for fluid flowing along the vector field (x³ + 2, y cos(6x)).

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: Find the flux of the vector field F(x, y, z) = 3xi+3yj+6zk across the portion of the surface z = 25…

A: The flux F across the surface is given by ϕ=∫σF·nds=∬RF·∇GdA

Q: Find the flux of the vector field F = (y, z, x) across the part of the plane z = 4+ x + 3y above the…

A: The surface integrals compute the flux of fields across surfaces,                   So,…

Q: Find the flux of the vector field F = (y, – z, x) across the part of the plane z = 4 + x + 2y above…

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Q: Let i = (3 – 2æyz - - xe cos y, yʻz, e² cos y) be the velocity field of a fluid. Compute the flux of…

A: Given that V=3-2xyz-xezcosy,y2z,ezcosy And the surface x+y2+z2=1 It is bounded. Then the flux is,…

Q: Calculate the vector field flux leaving F(x, y, z) = xi + yj + zi solid bounded by the surfaces x² +…

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

A: Given F→=x5+y  , y cos3x flux=∫cF→.n^  ds =∫∫Ddiv F dA

Q: Let divF(x,y, z) = 4x³ + 4y³ + 42³ , where F represents the flow of a vector field. Estimate the…

A: Concept Used:- Divergence Theorem Flux=∫∫∫∇.F dVwhere ∇.F is the divergence of FVolume of the…

Q: Compute the flux SS F.nds through the piece of the cylinder of radius 2, centered on the z-axis,…

A: The given surface is the cylinder of radius 2 centered on z-axis, where 0≤x, 0≤y and 0≤z≤4. The…

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

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Q: Let v = (x*z, 2 – 4xyz, xz) be the velocity field of a fluid. Compute the flux of v across the…

A: Given, v→=x2z,2-4xyz,xz225x2+16z2=y-202 Then, calculating div v→ div…

Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = ln(x + 5y + 6z). F(x, y, z) = =

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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: Is the two-dimensional vector field V = -2y 2x a possible velocity field x²+y² 'x² +y² for an…

A: For a two dimensional incompressible for a vector field V=ui+vj. ∂u∂x+∂v∂y=0

Q: The continuity equation for incompressible fluids states that the divergence of the velocity vector…

A: The given statement is The continuity equation for incompressible fluids states that the divergence…

Q: Let S be a simple closed surface in R' that encloses a solid region with volume equal to 5. Find the…

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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: Evaluate line integral of vector field F(z,9) = i+ 1- x? y2 2x along the curve x = cos(t), y =…

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Q: Find the flux of the vector field F = (y, - z, x) across the part of the plane z = 4+ 2x + 3y above…

A: The given vector field is, F→=y, -z,  x. The given part of the plane is z=4+2x+3y And the rectangle…

Q: Consider the line integral cos(x+2y)dx+2cos(x+2y)dy where S is a curve starting at (0,0) and ending…

A: We have to show that F is conservative. When F is given as F=cosx+2yi+2cosx+2yj For the conservative…

Q: Compute the flux of F = (5(z + 2), 5, 52) through the surface S given by y = ² + 2, with 0 0, z 2 0,…

A: We have to find flux.

Q: Find the work done by the force field F(x, y, z) = on particle that moves along the line segment…

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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 flow of the velocity field F =(x + y)i - (x2 + y2)j along each of the following paths…

A: The velocity field F=(x+y) i -x2+y2j along each of the following paths from 1,0 to -1,0 in the xy…

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: Find the work done in moving a particle in the force field F : 3x²i + (2xz - y)j + zk along the…

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Q: Evaluate the outward flux of the vector field F = (2x? – 5e" + sin(2²), –4ry + ln(x*), 5z – 3x²y°)…

A: Given

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

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Q: Find the upward flux of the vector field F(x,y,z)= (-y, x, 9) across the part of the hemisphere…

A: Consider the provided question,

Q: Calculate the work done in moving a particle in the force field F = 3x2î+ (2xz-y)ĵ+ z k along a…

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Q: Evaluate line integral of vector field F(x, y, z) = y sin(x²) i+ xyz j+ (x +y + z) k along line…

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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: Flow curves in the plane Let F(x, y) = ⟨ƒ(x, y), g(x, y)⟩ be defined on ℝ2.

A: The vector field Fx,y=fx,y,gx,y.

Q: Find the counterclockwise circulation and the outward flux of the field F = (-sin y)i + (x cos y)j…

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Q: F =(-y,x,z) across the solid sphere E %3D

A: we have to find the outward flux

Q: Find the outward flux of the field F = 3xz2 i + y j - z3 k across the surface of the solid in the…

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Q: Find the work done by the gradient vector field F(x, y, z) = V (e² + y² + z²) in %3| mov ing a…

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Q: Find the gradient vector field of f(x, y) = x° sin(9y). F(x, y) = x°i+ sin(9y)j F(x, y) = 9x°…

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Q: Without using a potential function, find the total work performed by the force field F(x, y) = (2x…

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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: Find the gradient vector field (F(r, y, z)) of f(x, y, z) = x² sin(3yz) F(z, y, 2) = (

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Q: The solid region E sits in the first octant, bounded above by the surface x² + y + z = 4. The vector…

A: Part (1):   The E is the solid region, now the volume is defined as dV = dz dy dx

Q: 6) Given the vector field F = (-x,-y,-z) '(x²+y2+z²)3/2! write the integral to calculate the outward…

A: First we parameterize field F then we calculate the Flux across the sphere using following formula:

Q: Compute the flux of the vector field F(x,y,z)=5xj+2yk→through the surface S, where S is the part of…

A: Flux problem...

Q: Let - (xz, 5 - 4xyz, xz) be the velocity field of a fluid. Compute the flux of i across the surface…

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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: A hot air balloon is shaped like a truncated sphere like the figure. The gases escape through the…

A: Given a hot air balloon has a truncated spherical shape. The velocity vector field  Vx,y,z=∇×Fx,y,z…

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: