Let F(x, y) = (5x², 2y ). Compute the flux 6 Fnds of F across the boundary of the triangle with vertices %3D (0,0), (2, 0), (0, 2) using the vector form of Green's Theorem. (Use symbolic notation and fractions where needed.)

Q: Find llFll and sketch several representative vectors in the vector field.

A:

Q: Compute the flux of the vector tield F(x, y, z) = 67 + 27 + 4k through the rectangular region with…

A:

Q: Find the outward flux of the vector field F = (x³, y³, z²) across the surface of the region that is…

A:

Q: Compute the flux of the vector field F(x, Y, z) = 6i + 5j + 3k through the rectangular region with…

A:

Q: 2. Find the r-component of the acceleration for the velocity field of a flow given by v(r, y, z,t) =…

A: NOTE: Hi! Thank you for the question. As per the honor code, we'll answer any one question in case…

Q: Find all points (a, b, c) on the sphere x2 + y2 + z2 = R2 where the vector field F = yz2i + xz2j +…

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: Compute the flux of the vector field F(x, y, z) = 5i + 5j+ 6k through the rectangular region with…

A: Given that, The vector field F→x,y,z=5i→+5j→+6k→ through the rectangular region with corners at…

Q: Find the curl of the vector field at the given point. F(x, y, z) = x2zi – 2xzj + yzk; (9, -9, 5)

A:

Q: Suppose given a function f(x, y, 2) of three variables, and a point (o, y0, 20) €R. Explain in your…

A:

Q: Find the flux of the vector fleldF = (x – y*)i+ 4yj + x'k out of the rectangular solid [0, 1] x [1,…

A:

Q: Find the flux of F = yi + 2zxj – z²k through the parabolic cylinder y = x2,0 < x < 1,0 < z< 4, in…

A:

Q: Find the gradient vector field of f(x, y, z) = x³y* + y° z³ + xz°.

A:

Q: Compute the curl, V x F, of the vector field. F(x, y, z) = 7xi + 9yj + 2zk curl(F) = -7xi – 5yj +…

A:

Q: for the vector field F = 2xz i+ (1 – 4xy² ) j led by the plane z = 6 – 2x² – 2y² and the plane

A: We have to find the following conditions according to the given information

Q: Find the gradient vector field of f(x, y, z) = e3r + 8y+6z

A:

Q: Find the gradient vector field of f(x, y, z) 5x+ 2y+8z

A:

Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = In(4x + y + 62) . F(1, y, 2) =

A: we have to find the gradient vector field of f

Q: Find the flux of the vector field F(x, y, z) = (yz, y°, xz) %3D across the surface S, where S is the…

A: Given

Q: Find the gradient vector field of f(r,y, z) = x*y* +y³z³ + æz°.

A: The gradient vector field of f(x,y,z) is given by ∇f=∂∂xf(x,y,z) i^+∂∂yf(x,y,z) j^+∂∂zf(x,y,z) k^…

Q: Calculate the flux of the vector field through the given surface.  F = 2i −3j + 6x2yk through the…

A:

Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = In(x + 3y + 4z) . F(2, y, 2) =

A:

Q: Compute the flux ∮∂D F • n ds of F(x, y) = {x3, yx2) across the unit square D using the Flux Form of…

A: Given the equation F(x, y) = (x3, yx2)  The closed surface is the unit square the limits of x and  y…

Q: Compute the flux of the vector field 1 F(x, y, z) = -k x² + y2 + 1 out of the conical container K =…

A:

Q: Find the outward flux of F = (4x +25y*,0,) 10z through the surface 25 4 9

A: Solution:-

Q: For the vector Ā =e - yzē, + xzē, and scalar function f(x,y, z) = x² sin(y) + y a) Obtain the…

A:

Q: Compute the net outward flux across the sphere r²+y²+z (r*, y², 2ª).

A:

Q: F = 2xi+2yj+3zk and σ is the portion of the cone z=sqrt(x^2+y^2) between the planes z=3 and z=5…

A:

Q: Find the curl of the vector field at the given point. F(x, y, z) = x²zi – 2xzj + yzk; (9, -9, 9)

A:

Q: Find the gradient vector field (F(x, y, z)) of f(z, y, z) = e®2z + $y + 2= F(1, y, z) =

A:

Q: Find a vector function that represents the curve of intersection of the paraboloid z=7x^2+5y^2 and…

A:

Q: Find the flux of the vector field F = 2 i +3 j – 2 k through the rectangle with corners at (1,0,0),…

A:

Q: Let F(x, y) = (4 cos(y), 2 sin(y)). Compute the flux $ F - n ds of F across the boundary of the…

A: By using Green Theorem, ∮F.n ds…

Q: Find the gradient vector field of flx, y, z)=x'y+y%z4+xz6 O V flx, y,z)= (3x?y+z°, 5x³y³ + 4y*zA,…

A:

Q: Match the vector field with its graph. F(x, y) = xi + 3yj 5 5 5 -5+ -5+

A:

Q: - 25y,0,) 10- through the surface Find the outward flux of F =1.

A:

Q: Find the curl of the vector field at the given point. F(x, y, z) = x²zi – 2xzj + yzk; (9, -9, 7)

A: The curl of a vector field F is defined as  ∇×F=i^j^k^∂∂x∂∂y∂∂zf1f2f3

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

A:

Q: Find the flux of the curl of the vector field F(x, y, z) = [(z2 −4) cos(xyz), 3x+y,(z2 −4) sin(ex…

A: Surface is oriented outwards. And -2 &lt;= z &lt;= 2

Q: Let F(x, y) = (2x + y³, 2y – x4). Compute the flux oF. F.n ds of F across the circle of radius 5…

A: Given function can be written as, Region bounded by the circle of equation,

Q: Suppose that in a certain region of space the electric potential V is given by V (x, y, z) = 8x ^…

A:

Q: For a scalar function f(x, y, z) = x? + 3y2 + 2z?, the gradient at the point P(1, 2, - 1) is

A:

Q: Let T(x, y,z) = x² + y² + z² represent temperature and let the 'flow' of the heat be given by the…

A: The flux of heat out of the given sphere is given by: Here is the vector field and is the surface…

Q: Find the circulation and onotward flux of F = aroud and u2 %3D across the ellipee +- Circulation…

A: Given the vector field is F→=-y,x. Find the circulation and outward flux of F→ around and across the…

Q: Let ƒ(x,y) = x²e"9. Give the unit vector that points in the direction of maximum increase along the…

A: Given that: f(x,y)=x2exy&amp; the point is (x,y)=(1,1)

Q: Find the Duf of f(x, y,z) = z In(x+ y) + e3y on the vector u = 3i -j-4k at Po(1,0,1)

A:

Q: Find the gradient vector field of f(x, y, z) = e® 5x+2y+8z

A:

Q: Find the curl of the vector field at the given point. F(x, y, z) = x²zi - 2xzj+yzk; (9,-9, 9) curl F…

A:

Q: Compute the flux of the vector fieldi = -3 i+ 4j- 2 k through the rectangular region shown below,…

A:

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: