Consider the surface S given by y=8−x2−z2, y≥0 oriented in the positive direction of the y axis. be the field F(x;y;z)=(−11y+ln(z2+1);68x2+y+z17;ey+21z+arctg(x+17)). What is the value of the flux of F through S?

Q: Solve the line integral with the vector field F (x, y) = (xy, seny) and the parametrization x (t) =…

A: The solution is given as

Q: Given the vector field E = 4zy² cos 2xa, +2zy sin 2xay + y² sin 2xa₂ for the region [x], [y], and…

A: The general form of a vector-valued function in three dimensions is given by F=F1ax+F2ay+F3az, where…

Q: What is the (numerical) value of the 3-form field (x² + y + z?) dx A dy A dz on the ordered triple…

A:

Q: The following figure shows a half-cylindrical closed surface, defined by the cylinder x² + y² = 4…

A: As per the question we have to find the total flux of the vector field V through the flat top…

Q: Find the counterclockwise circulation and outward flux of the field F = xyi + 4yʻj around and over…

A:

Q: Let C be the ellipse in which the plane 2x + 3y - z = 0 meets the cylinder x2 + y2 = 12. Show,…

A:

Q: Find the outward (normal away from the z-axis) flux of the field F = y2î + xzî – k across the cone z…

A: In this, we find the flux of the given function across the given cone z=3√(x^2+y^2).

Q: Compute the work done by the force field F =< x² + 2y, 4x + y° where C is the boundary of the region…

A:

Q: Consider the gradient vector field V = Vg of the function g(x, y) = x² + y² + e*²-y´ cos(2xy).…

A: We will use the Divergence Theorem in 2D to solve the given flux integral.

Q: Given a scalar field f(x, y)= xy + x² and C is the curve defined by the equation y =-x². Calculate F…

A: Given scalar field  fx,y=xy+x2 and C is the curve defined by the equation y=-x2.

Q: I. Use the Divergence Theorem: (² F. ds = Ifv. Fav SSS E = to calculate the flux of the field F…

A: Divergence theorem uses  Double integration converts into triple integration.

Q: 3. Consider the vector field F = (y, -x, x² + y?). Find the flux of F through S if: 9– x2 – y? above…

A:

Q: Find the flow of the field F = V(x²ze) once around the ellipse C in which the plane x + y + z = 1…

A: Consider the given field, F=∇x2zey around the ellipse in which the plane x+y+z=1intersects the…

Q: Consider the surface integral of the vector field F = [xy?, 3x², x²z] over a surface S, where S…

A: We will use of gauss divergence theorem

Q: Use the divergence theorem to find the outward flux F ndS of the vector field F = cos(8y + 8z) i + 4…

A: …

Q: Let f(x, y, z) = (x² + y² + z²)¬/2. Show that the clockwise circulation of the field F = Vf around…

A: Calculate: F=∇f   =∂∂xx2+y2+z2-1/2,∂∂yx2+y2+z2-1/2,∂∂zx2+y2+z2-1/2   =-x2+y2+z2-3/2x,y,z

Q: Find the work done by the following force field F(x, y) = 3(v+ 2)'0i + 30x (v + 2)°j %3D in moving…

A: The work done is a scalar quantity, which is equal to the dot product of force applied to the body…

Q: Find the net outward flux of the fieldF = ⟨bz - cy, cx - az, ay - bx⟩ across any smooth closed…

A: The vector field is F=bz-cy,cx-az,ay-bx where a, b, and c are constants.

Q: Let W be the work done by the forced field F(x,y,z) =yi+zj in order to move a particle along the…

A: The formula for work done is W=∫F.dr.

Q: Find the flux of the field F = x i + y j + z k outward through the portion of the cylinder x2 + y2 =…

A:

Q: Calculate the flux of the field F across the closed plane curve C where F=xi+ vj and the curve is…

A:

Q: Find the flux of the field F =-x i - y j + z2 k outward (normal away from the z-axis) through the…

A:

Q: Find the counterclockwise circulation and outward flux of the field F = xyi + y2j around and over…

A: Theorem used:

Q: Find the gradient field F = Vq for the potential function p below. P(x,y,z) = In (x² + 3y? + 4z²)…

A: Find the gradient field for the potential function φφx,y,z=lnx2+3y2+4z2

Q: Suppose that v = (0, y, 0) is the velocity field of a fluid and let S be the surface defined by y =…

A: Given, the velocity field v→=0, y, 0and S be the surface defined by y=x2+z2, with y≤1.

Q: Use the Divergence Theorem to compute the net outward flux of the field F = (4x, y, - 2z) across the…

A:

Q: Calculate the flow/ line integral in the field F along the path C. F=y2i+zj+xk; C is the curve r(t)…

A:

Q: 12. Use a parameterization to find the flux across the surface cut from the cylinder x? + y? = 1 by…

A: Image is attached with detailed explanation.

Q: Compute the work done by the force field F = , 0 <t<1

A:

Q: Find the outward flux of the field F = 6xyi + 4yzj + 2xzk across the surface of the cube cut from…

A:

Q: Given the vector field F (x, y, z) = 2xyi + (x² + z?) j +2yzk a. Verify that the vector field…

A: Ans :A)

Q: Consider the vector field F (xr, y) = (2y - 12r*y", 4ry – 9r*y?) is conservative. (a) Show that F…

A:

Q: Is the force field F (x, y) = x³i + sec(x+ xy)j conservative?

A:

Q: Given the radial field (x, y, z) (x² + y² + z²)³/2 |r|3/2 Find the outward flux across the sphere of…

A: Image is attached with detailed solution.

Q: Use the Divergence Theorem tO compute the flux oT the field 1 25x cos z, 6xy ) out of the region E…

A: Given,       1z2+y2arc tanxz2+y2, 25xcosz, 6xy out of the region E that lies between the spheres…

Q: Find the circulation of the fieldF = (x2 - y)i + 4zj + rk around V? + y?, the curve C in which the…

A: Field F→=x2-y i^+4z j^+ x2 k^Surface S is cone z=x2+y2at plane z=22=x2+y2x2+y2=4So, f=x2+y2-4

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

A:

Q: 2. Find the surface integral SS. ř · dS for the vector field F = xyî + yzĵ + zxk where S is part of…

A: To find Surface integral ∬SF→·dS→: The Surface S is of the form z=g(x,y) and is oriented upwards.…

Q: Find the potential function for the following vector field F(x, y, z) = (2x cos y – 2z3) î + (29+…

A: F = &lt;2xcosy-2z3, 29+2ye2-x2siny, y2e2-6xz&gt; To check F¯ is conservative  not we have t check…

Q: For the force field F = (2xe¬Y, (2y – x²e¬y)) in some open connected region containing P(1, 0) and…

A: Given that, F=2xe-y, 2y-x2e-y If F is conservative then ∂2F∂x∂y-∂2F∂y∂x=0

Q: (b) A particle moves once around the circular path with radius of 1. Using Greens' Theorem, compute…

A: see below the answer

Q: 6) Given a hemisphere bounded by S,: z = /25 – x2 – y? and S2:z = 0.Find the outward flux of the…

A: The flux of a vector-valued function F(x,y,z)=F1i+F2j+F3k over a surface S is defined as ∫SFdS.…

Q: Find the outward flux of the field F = 2xy i + 2yz j + 2xz k across the surface of the cube cut from…

A:

Q: Use the Divergence Theorem to compute the net outward flux of the field F = (4x, 3y, - 3z) across…

A: Let's find.

Q: Why does a two-dimensional vector field with zero divergence on a region have zero outward flux…

A:

Q: Consider the force field F(x, y, z) = xi + yj + zk. Compute the work done in moving a particle along…

A:

Q: Find the outward flux of the given field across the given cardioid. F=(3ry-)i+(e*+5tan-'y)j 1+y?…

A: The outward flux of given field is given by using greens theorem.

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

A:

Q: Calculate the flux of the field F across the closed plane curve C where F= xi+ vj and the curve is…

A:

Q: Find the gradient field of the function f(x.y.z) = In 4x + 4y2 +z?. ..... The gradient field is Vf=…

A: Gradient of F is given by