Verify Stoke's Theorem fF-dr = [[ V×F•d$ by evaluating both side of the equation. The field is given by F(x,y,z)=(-y+z)i+(x-z)j+(x-y)k and the surface S is given by ーy

Q: Use the Divergence Theorem to compute the net outward flux of the field F = (2x,3y,2z) across the…

A:

Q: gral of the field F(x,y,z) = 7xi + 7yj + 7zk across the portion of the sphere x2 +y² +z? = a? in the…

A:

Q: Verify Green's theorem F - dř = P(r, u) dzr + Q(z, y) dy = | | {0.Q(z, y) – ,P(x, y)} d.ædy, %3D for…

A:

Q: (a) Show that the vector field F(x, y) = (x², y²) is conser- vative, then find the potential…

A:

Q: verify the conclusion of Green’s Theorem by evaluatingboth sides of Equations (3) and (4) for the…

A: The vector field is F = -yi+xj. The domain of integration is the disk R = x2 + y2 ≤ a2 and its…

Q: 3. Use Gauss' theorem to evaluate IL (* + y + 2) do, where E is the surface of the ball x? + y? + z²…

A:

Q: 8. Let F(x, y) =. x2+y2' x²+y2 a. Evaluate the integral S. F ·dš where c(t) =; 0<t< 2n. У х b. Show…

A:

Q: Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F and…

A: To use Green's theorem to find the counterclockwise circulation and outward flux for the field F and…

Q: Use Green's theorem to find the counter-clockwise circulation for the field V = 2xyi – 3x°yj, where…

A:

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: use Green’s Theorem to find the counterclockwisecirculation and outward flux for the field F and…

A: Given that, F=x2+4yi+x+y2j Bounded by x=0, x=1, y=0, y=1

Q: Consider the scalar functions 1(2,9) = arctan(). %3D (2, y, =) = sin(z)e In(2), and the vector field…

A: Hormonic function, gradient of the vector  function

Q: Use the Divergence Theorem to compute the net outward flux of the following field across the given…

A: We will find out the required value.

Q: Find the circulation and flux for the field F(x, y) = (y – 6x² , x+y²) around and across the…

A: Given data field Fx,y=y-6x2,x+y2 around and across the triangle given by y=0, y=x and x=1.

Q: F = (3x+2y,4y-5x), C is the rectangle 0SxS3, 0<y<l.

A: Fg

Q: Consider the scalar field f(x,y, z) = 2x² – y² + 3z? Determine V Vfl(1.-1.1) II. VxVfl1.-1.1)

A: Here, the given function is: f(x,y,z)=2x2-y2+3z2

Q: Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F = (4x…

A:

Q: use Green’s Theorem to find the counterclockwisecirculation and outward flux for the field F and the…

A: Green theorem (Outward flux): ∮CF.nds=∫∫R∂M∂x+∂N∂ydxdy Green's theorem (circulation form):…

Q: Q4// A// Evaluate both side of divergence theorem for the field D=2xy.x^+x².y^ C/m2 and the…

A:

Q: 3. Find a field F(x,y) = (M(x, y), N(x, y)) in the xy-plane with the property that at each point

A: Let f = xi + yj  Then F = ff = unit vector = xi+yjx2+y2

Q: ) Verify Stokes' Theorem for the hemisphere S : x²+y²+z² = 9, z > 0, %3D its bounding circle C: 1?…

A: Stokes theorem  If a smooth surface S bounded by closed ,smooth boundary curve C oriented in…

Q: use Green’s Theorem to find the counterclockwisecirculation and outward flux for the field F and…

A: Solution:GivenF=(xy+y2)i+(x-y)jFormula:Outward Flux=∫F.n ds=∬ ∂M∂x+∂N∂y dx dyCounter Clockwise…

Q: Use the Divergence Theorem to compute the net outward flux of the following field across the given…

A: The equation of given surface is F=y-x,5z-2y,2y-5x. Here surface is the faces of cube with limits…

Q: (a) Express the field D = (x² + y²)-'(xa, + ya,) in cylindrical components and cylindrical…

A: Express D=x2+y2-1xax+yay in cylindrical components and cylindrical variables.

Q: Verify Stoke's Theorem for the vector field F = (2y + 2) i + (x –2)ĵ+(y – x) . over the portion of…

A: The formula for stokes theorem is, ∫Pdx+Qdy+Rdz=∬∇×F.ndS Given that, F=(2y+z)i+(x-z)j+(y-x)k Then,…

Q: plem #4: Use the divergence theorem to find the outward flux || F.n dS of the vector field F =…

A:

Q: S is a closed surface made from two surfaces, S1 and S2. The first surface S1 can be described by a…

A: Let S be the surface bounded by a circular paraboloid z=2-x2-y2 and z=x2+y2. Given the vector field…

Q: ation of the field F(x, y, z) = (x, y, xy) along the boundary of the surface obtained as the x2 y…

A:

Q: Consider the optimisation of the scalar field f(r,y) = (y+2)(y² + 4y+z²) on the set D = {(x,y) | x²…

A: Given function is, fx,y=y+2y2+4y+x2 Differentiating the given function partially,…

Q: Use the method of isoclines to sketch the slope field for y =x +y-1, then draw the solutions…

A: In this question, we sketch the direction field for the differential equation y'=y+x2-1 To draw the…

Q: 1. The vector field F(x,y) = y i+ + y? j x² + x² + y? Is an example of a vector field that does not…

A: This question is from vectors.

Q: (x - x²y, xy²) along the boundary of D = {(x, y) E R : x² + (y – 1)² < 1}, counterclockwise…

A: Circulation of the vector field

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: Is it true or false that the vector field below is conservative? -y -X -Z F(x, y, z) = (x² + y? +…

A: Given,                              =-(xi+yj+zk)(x2+y2+z2)32

Q: Find a field F = M(x, y)i + N(x, y)j in the xy-plane with the property that at each point (x, y) ≠…

A:

Q: 1. The vector field F(x, y) = -24 yi i + -y x² + x² + y2 Is an example of a vector field that does…

A: Line integrals of a vector fields is given by  If F(x,y)=P(x,y), Q(x,y) is a vector field , and the…

Q: 1. Use Green's Theorem to find the counterclockwise circulation of the field F around the curve C as…

A:

Q: Verify the Green's Theorem for the fields F = 2ryi - 2.ry²j. Take the domain of the integration in…

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: Find a field F = M(x, y)i + N(x, y)j in the xy-plane with the property that at each point (x, y) ≠…

A:

Q: Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F and…

A:

Q: 1) Verify Stoke's Theorem for the vector field F(x, y, z) = 2zı + 3x) + 5yk, taking S to be the…

A: Let F be a vector field and C be the boundary of the surface S. The Stoke's theorem states that…

Q: 12. Find the flux of the field F(x, y, z) = 4xi + 4yj + 2k outward (away from z-axis) through the…

A:

Q: Verify the conclusion of Green's Theorem by evaluating both sides of each of the two forms of…

A: As we know that Green's Theorem turns a line integral into a double integral. It helps to simplify…

Q: 2. (a) Let r = (x, Y, z) and r = ||r||.  Assuming r cannot equal 0, is there a value of p for which…

A: This is a problem of vector calculus.

Q: Let the curve C be the boundary of the triangle cut from the plane 2x +y +3z =3 by the first octant,…

A: We will use Stokes Theorem to find the circulation: ∮CF⋅dr=∬S(∇×F)⋅n^dS

Q: Use the Divergence Theorem to evaluate the flux of the field F(x, y, z) = (2e*+y, 2e*+z, 2e*+y)…

A:

Q: Q4: A- Find the flux of the field F = i+j+k √√x² + y² +2² bounded below by the plane z-0 and bounded…

A:

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.