Let divF(x, y, z) = 4x³ + 4y3 +4z³, where F represents the flow of a vector field. Estimate the flux of F out of a small sphere of radius 0.1 centered at the point (1,1,1). Please express your answer as a decimal rounded to the nearest 0.0001.

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

A:

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

A:

Q: Find the total work done in moving particle in a free field given by F = 3xyi – 5zj + 10xk along the…

A: The total work done by a vector field, F→ moving a particle along the curve r→(t) is calculated…

Q: What is the flux of the vector field F(x, y, z) = [9y, 2xy, 4yz + 234xy] through the unit cube [0,1]…

A: We find the flux of the vector field F(x, y, z) = [9y, 2xy, 4yz + 234xy] through the unit cube [0,1]…

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: Calculate the flux of the vector field F(x, y, z) = 5i + 2j+ zk through the closed circular cylinder…

A:

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

A:

Q: Find the flux through through the boundary of the rectangle 0 < x < 6,0 < y < 3| for fluid flowing…

A:

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

A:

Q: Find the net flux of the vector field F(x, y, z) = (2x, - 2y, z) over the closed surface enclosed by…

A:

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

A:

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: 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: 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…

A:

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

A:

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

A:

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

A:

Q: Find the work done by the force field F(x, y) = (x^2-lny)i + (e^x^2+y^2)j on a particle that travels…

A:

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

A:

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

A:

Q: Why is the upward flux of a vertical vector field F = ⟨0, 0, 1⟩ across a surface equal to the area…

A:

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

A:

Q: How is the circulation of a vector field on a closed smooth orientedcurve calculated?

A:

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

A:

Q: Find the flux of the vector field F = across the part of the plane | z = 2 + x + 3y above the…

A: Given: F¯=y,-z,x z=2+x+3y The formula to determine the flux is: ∫S∫F·nds=∫D∫(-agx-bgy+c)dA Here…

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

A:

Q: If F = xy i+ (x² + 4yz)j+ 2y²z k. Evaluate the line integral between A (0, 2, 0) and B (3, 6, 1).…

A: Given, F= xy i +(x²+4yz)j +2y²zk r0= (0,2,0) r1= (3,6,1)

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

A: Given F=xi+yj+zk ,-3&lt;=x&lt;=3,r=1

Q: Find the gradient vector field (F(r, y, z)) of f(x, y, z) 5zy ze F(1, Y, 2) =

A: Gradient vector field F→(x,y,z) of a function f(x,y,z) is given by,…

Q: Find the gradient vector field (F(r, y, z)) of f(x, y, z) = z sin(ry). F(2, y, z) = (

A: Given function is  f(x,y,z) = z^2 sin(xy)

Q: Compute the flux of the vector field F(x, y, z) = 3xj + 3yk through the surface S, where S is the…

A:

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

A:

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

A:

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: 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…

A:

Q: Compute the flux of the vector field F(x, y, z) = 41 + 45 + Sk through the rectangular region with…

A:

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…

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

A:

Q: Find the flux of the constant vector field i= -4 i-5j-2 k through a square plate of area 16 in the…

A: The objective is to find the flux.

Q: Compute the flux of the vector field F = 2xi + 2yj through the surface S, which is the part of the…

A:

Q: Find the flux of the vector field (y, x, z) in the negative z direction through the part of the…

A:

Q: Find the work done by the force field F(z,y,2) = Ir+v,rs, y+:) on a particle that moves along the…

A:

Q: Calculate the vector field flow F(x,y,z)=(x2,y2,z2)  through S, where S is a flat surface x+y=2,…

A: To evaluate ∫∫SF.dS: By the divergence theorem, ∫∫SF.dS=∫∫∫Ediv F dV where E is the solid enclosed…

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: