Compute the flux of the vector field F = xi + yj through the surface S, which is the part of the surface z = 25 - (x2 + y?) above the disk of radius 5 centered at the origin, oriented upward. flux =

Q: What is the line integral of a vector field?

A: To define the line integral of a vector field.

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: The gradient of a scalar field V is a vector that represents both magnitude and the direction of the…

A: Given : The gradient of a scaler field V is a vector that represents both magnitude and direction…

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

A:

Q: 5) By using Stokes' Theorem, find the work performed by the vector field Ex² + 4xy³i+ y²xk,on a…

A:

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

A:

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

A:

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

A:

Q: A pasticle depicts ts motion in Such a ulay at time ť is P=xîty where, 2= acoskt and y=bsinkt. that…

A:

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

A: Solution is given below

Q: Sketch a few representative vectors of vector field F = ⟨0, 1⟩along the line y = 2.

A:

Q: Sketch the vector field in the plane along with its horizonta and Vertical a representative…

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: Calculate the outward flux of the vector field F→ (x,y,z) = 2xi→ + yi→ +zk→ along the boundary of…

A:

Q: Calculate the outward flux of the vector field F→ (x,y,z) = 2xi→ + yi→ +zk→ along the boundary of…

A:

Q: Calculate the scalar curl of the vector field. F(x, у) %3D yi — хj

A: To calculate the scalar curl of the vector field.F(x,y)=yi^-xj^

Q: Compute the flux of the vector field F = 3xzi + 5yk through the surface S, which is the hemisphere…

A: The flux is given byflux =∫S  F→·n^  dShere,n^=x, y, zx2+y2+z2 in spherical coordinatesF→·n^ =x, y,…

Q: 1. Verify the Divergence Theorem in the calculation of the flux of the vector field: through the…

A:

Q: a vector field ū in spherical courdinate system Express t in the Cartesian coordinate system. U= -…

A: In spherical coordinate system, u→=urr^+uθθ^+uϕϕ^u→=-cosϕ2r+sinϕ2ϕ^ The Cartesian coordinate system,…

Q: Find work done by a force field F= in moving object along the line segment from the point (2,1,3) to…

A: Given, F=1y-x,xyz,ex2+y2+z2

Q: The work done by a force field in a moving particle around a closed path is zero.

A:

Q: Estimate the circulation of a vector field F around a circle of radius R = 0.1, assuming that curl2…

A: It is known that ∮c2F·ds-∮c1F·ds=∬DcurlFdxdy where c1 and c2 are circles of radius 0 and 0.1,…

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: Compute the flux of the vector field F 3yi + 4j – 3xzk through the surface S, which is the surface y…

A: We apply formula for finding this

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

A:

Q: Find the level surfaces of the scalar field U = ear, where the vectors are: a = (2, 2, 1) and r = =…

A:

Q: Sketch a two-dimensional vector field that has zero curl everywherein the plane.

A:

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: Find potential 9, for the conservati ve field the vector F (x, y,a)- (2xe? - y*sinx) î * sinx) î +…

A:

Q: Compute the flux of the vector field F(x,y, z) = yi– 2xyk over the triangle T with vertices (1,0,0),…

A:

Q: Find the flux of F = yi+2xj-3zk through the unit disk centered at (0,3,0) in the plane y=3, oriented…

A: Given F→=yi^+2xj^-3zk^ through the unit disk centered at 0,3,0 in the plane y=3 oriented towards the…

Q: Find the curl of the vector field F=(2xyz, -xy,-z^2)

A:

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

A:

Q: Sketch the vector field of F = (-y,0) in the xy-plane. %3D

A:

Q: Find the outward flux Of The field F across the boundary surface of the solid.

A:

Q: Calculate the flux of the vector field F 6i + 2x2j- 2k, through the square of side 4 in the plane y…

A: Given vector field is F→=6i→+2x2j→−2k→ We have to find the flux of the given vector field. The flux…

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…

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:

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: Compute the flux of the vector field F = 2xzi + 5yk through the surface S, which is the hemisphere…

A: In this question, we find the flux for the given vector field across the surface of the hemisphere.

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: 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: a) What is the flux of the constant vector field F = (a, b, c) through any closed surface? b) Is the…

A: Given That : A constant  vector field F=a,b,c and F=x3,y3,z3 To Find : i) The Flux through of…

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: