A hot air balloon is shaped like a truncated sphere like the figure. The gases escape through the porous coating whose vector field isF(x, y, z) = (-3y,4x,0). If the radius of the untruncated sphere is 12, calculate the volumetric flux of the gases through the surface

Q: Find an equation of the plane through the origin such that the circulation of the field F = zi + xj…

A: Given the circulation of the field F = zi + xj + yk along the intersection of the planeand the…

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: A quarter cylinder of radius 2 and height 5 exists as shown in the figure below. Surface integral of…

A: Given that the quarter cylinder is of radius 2 and height 5. This means that the radical distance…

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…

A:

Q: Use the Green's theorem to find the work done by a force F(x, y) = on a particle that moves…

A: find explanation below

Q: Find the work done by the force field F(x,y) = 4yi + 2xj in moving a particle along a circle x2 + y2…

A: We follow

Q: Compute the flux of the vector field F = xi + 5yj through the cylindrical surface as shown in the…

A:

Q: calculate the flux of the vector field through the given surface F = zi + (y −1)j + 5xk, and S is…

A:

Q: Find a flux of the vector field ä(M)= (2z– y)i +6xj + yk through a part of the surface S: x - 4y+2z…

A: Given, a→M=2z-yi^ +6x j^ +y k^ Also given surface is, S: x-4y+2z-4=0 So the normal vector to the…

Q: Consider the vector field F = (2y +1,y, –2²) along with the paraboloid surface z = r² + y. Find the…

A: Circulation is the line integral of the vector field along the curve

Q: Use the Divergence Theorem to calculate the flux of the vector fields F1 = -8zi + 8ak and F2 = -8zi…

A: Given vector fields are F1→=−8zi^+8xk^ and F2→=−8zi^+y+2j^+8xk^ We have to calculate the flux of the…

Q: Find the flux of the vector field F(x, y, z) = ri across the outward oriented sphere %3D

A: In this question, we find the flux of the vector field f(x,y,z)=xi The surface is a sphere of radius…

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: The flow of a gas with a density of δ = 0.001 kg/m2 over the closed curve r(t) = (-sin t)i + (cos…

A: Given that

Q: Find the work done by the force field F(x,y) = xyi + y²j around the circle (Clockwise direction)…

A:

Q: Find the work done by the force field F(r, y) = x²i+ xyj on a particle that moves once around the…

A:

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: 4. Compute the outward flux of the vector field F(r) = (r, y', =') through the unit sphere r² + y² +…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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

A:

Q: Let = (xz, 2 4xyz, rz) be the velocity field of a fluid. Compute the flux of i across the surface…

A: Flux of velocity about a surface is the measure of amount of velocity of fluid hitting the surface.…

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

A:

Q: Find the conservative vector field for the potential function by finding its gradient. 9 (r,y, 2) =…

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Q: Compute the flux of the vector field F(x, y, z) = 2xj+ 4yk through the surface S, where S is the…

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: 2. Let C be the curve of intersection between the cylinder r2+y = 1 and r-2y+z = 0, oriented…

A: The complete solutions are given below

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: Find the work done by the vector field ⟨4x+yx,x2+6⟩ on a particle moving along the boundary of the…

A:

Q: Find the work done by the force field F(x,y) = x²i – xy²j around the circle (Clockwise direction)…

A:

Q: Find the outward flux of the vector field F = (x°, y°, z²) across the surface of the region that is…

A:

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

A:

Q: Find the work done by the force field F(x,y) = (-y,x) in moving an object along the path y = V4 – x2…

A: Given force field Fx,y=−y,x and object moving along the path y=4−x2  from 2,0 to -2,0. We have to…

Q: Find the flux of the vector field F(x, y, 2) = yj + k where o is the portion of the cone =' = 4(x² +…

A:

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

A:

Q: 2的 在年

A:

Q: Find the work done by the force field F(x, y) = x* +y? on a particle that moves along the quarter…

A:

Q: The vector field F = (, i, 2²) has a scalar potential in the region R= {(x, y, 2) E R³ : r² + y? >…

A: We find out true or false.

Q: The circle of radius a centered at the origin, and the vector field F = -i + yj.

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: For what values of a and d does the vector field F = ⟨ax, dy⟩ have zero flux across the unit circle…

A: A vector field, have zero flux. Across the unit-circle centered at the origin and oriented…

Q: Is the vector field F = [yze"yz, xze#yz, xye"yz] conservative? %3D O Yes O No

A: The vector field is conservative if ∇×F→=0 therefore, find ∇×F→ ∇×F→=ijk∂∂x∂∂y∂∂zyzexyzxzexyzxyexyz

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

A: The solution is given below

Q: he temperature in a temperature-controlled chamber is given by a 3D scalar field Tlx, y, 2) = Ax®yz…

A: Note- Since you have asked multiple questions, we will solve the first question for you. If you want…

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

A:

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

A:

Q: Find the flux of the force field F(x, y, 2) = (2, x + r², 1 – a² + y) across the portion of the…

A:

Q: Find the flux of the vector field F = [x,y,z] through the oriented surface S: z= x^2 + y^2, z <= 1…

A:

Q: Find the work done by the force field F(x,y) = x²yi + y²j around the circle with radius 2.

A:

Q: A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at…

A: A sphere of radius 1 is x2 + y2 + z2 = 1. Parametrize the sphere as follows:…

Q: Sketch the vector field F(x, y) = 3i + 5j and calculate the flux then the circulation of F| along…

A: (1) The equation of line segment is: x+3-2+3=y-1-5-1=tx+3=t⇒x=t-3y-1=-6t⇒y=-6t+1 The curve…

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: