7. Use the Divergence Theorem step by step including sketch of the solid shape to find the flux f SF ndS out of the given sold. If F = (2x², y°, yz) and o is the ball 1² + y? + z? <4. %3D

Q: 4. Let D = y²z°ãx + 2xyz³ ày + 3xy?z² đz pC/m² in free space. a. Find the total electric flux…

A: Solution:-

Q: Let S be the surface of the solid hemispherical region above the plane z = 0 and below the sphere…

A:

Q: 46.) (a) Parameterize S the 1st octant portion of the cylinder y² + z² = 9, 0 <x<4 (b) Use above to…

A: y2+z2  0≤x≤4x=uy=3 cos vz=3 sin vparameterise the surface ass(u,v)=(u   3 cos v   3 sin v)0≤u≤4,…

Q: 6. In each case, compute the flux (x°, y°, z³). W- (a) F(x, y, z) = and a? + y? + 2² = 9 anc…

A: Use divergence theorem

Q: And the Vorume of the solid gererated by rovolung the reglon boundid by the given ines't (urves…

A:

Q: Find the Uolume of the Solid under de the 2:1+ xy2 & the a Gove te region Sorface en cased by y= x²…

A: Consider the given surface z=1+x2 y2 Since, y=1 and y=x2 Then, x=±1 That is, -1≤x≤1  and    x2≤y≤1

Q: If f (0)=f'(2), find the difference c-2d for the following cubic spline S(x) with clamped boundary…

A:

Q: A flux is given by F = x²y³ i – xy j. Verify the Green's theorem for the region bounded by the…

A:

Q: 14 13 12 11 10 9 8 7 4- 3- 2- Find the volume of the solid S obtained by rotating R about the line y…

A: Here R is the region bounded by y=9x,x=2.5 and the x-axis i.e.(y=0)and the solid…

Q: The volume of the solid generated by the curve y = e, between z = 2 and z = 6 about the z- azis is

A: Given: The curve is y=e-x   To determine: The volume of the solid generated by the curve y=e-x…

Q: 19. Evaluate II \xyz| dx dy dz over the solid ellipsoid x² y2 < 1. a? (Hint: Let x = au, y = bv, and…

A:

Q: Evaluate the triple integral. 16.  ∫∫∫T  xz dV , where T is the solid tetrahedron with vertices…

A: The solid tetrahedron T with vertices 0, 0, 0, (1, 0, 1), (0, 1 ,1) and 0,0, 1. The solid region is…

Q: Find a solution to the following Dirichlet problem for a half disk. Zu 1 du 1 Pu = 0, 0<r<1, 0<0<n…

A:

Q: 1. Evaluate SSSE z dV where E lies between the spheres x2 + y? + 2² = 1 and a? + y² + z² = 4 in the…

A: Given, ∫∫∫EzdV where E lies between the spheres x2+y2+z2=1 and x2+y2+z2=4 in the first octant.

Q: 12. Consider the lamina covering the regon x442-2,0EXET WH aunsity functionplx,u) = - 5x + 2u+5. 1ts…

A:

Q: Find a solution to the Dirichlet boundary value problem for a disk. 2u 1 du 1 Pu = 0, r dr r2 a02 +…

A: Answer is mentioned below

Q: |s, is calculated by the thorem of divergence of the full range straight flux F(x, y, z) = 2xỉ + yj…

A: Answer  Flux = 2π

Q: . Solve the following Dirichlet problem on a disk V²U(r,0) = 0, 0<r<1, -asosa U(1,0) = 0, - <o< T.

A: This question based on Dirichlet problem on a disk from partial differential equation.

Q: Evaluate ſ[ſp z dV where E lies between the spheres x² + y² + z² = 1 and + y*¨+ 2² = 4 in the first…

A: Solution:

Q: 5) S, is calculated by the thorem of divergence of the full range straight flux F(x, y, z) = 2xi +…

A:

Q: 3. Calculate the circulation of A=p coso ap+ z sino a: around the edge L of the wedge defined by 0≤p…

A: Here we have given that, A=ρcosϕaρ+zsinϕaz and wedge is defined as, 0≤ρ≤2, 0°⩽ϕ≤60° and z=0 as shown…

Q: Evaluate the triple integral xy dV where E is the solid tetrahedon with vertices (0, 0,0), (6, 0,…

A:

Q: (10) Find the work done by F = -yi+ xj+ zk on a particle moving along the helix r(t) = where 0 < t <…

A:

Q: Q (3): Given the field D = p sin 0.5o a, + p cos 0.50 a, +pa, C/m, evaluate both sides of the…

A: According to the formula of divergence theorem, I=∫∫V∫div V dv=∫∫∫∂Vρ∂ρ+∂Vϕ∂ϕ+∂Vz∂zdρdϕdz Now…

Q: 5) S, is calculated by the thorem of divergence of the full range straight flux F(x, y, z) = 2xỉ +…

A: Given: F(x, y, z) = 2xi + y + zk along, almost every where with z = x² + y² paraboid and z = 2x…

Q: If D = 15p sin 2øa, + 10p² cos 2pa, C/m², evaluate both sides of the divergence theorem for the…

A:

Q: 1. Evaluate fL, 2 dv where E lies below the plane z = 2+ 3x + 4y and above the region in the…

A:

Q: 2. Set up the triple integral f[Sp I+y+zdV, where D is the tetrahedron with vertices at (0,0,0),…

A:

Q: 1. Use the cylindrical coordinate to evaluate ff. x² + y²dV _where G is the solid bounded by…

A:

Q: FD is the region in the first octant between the spheres x2+y2+z²=1 and x2+y2+z²=16, then dV =

A: Converting into spherical coordinates

Q: 5) S, is calculated by the thorem of divergence of the full range straight flux F(x, y, z) = 2xỉ +…

A:

Q: Given the field D = 6sin 0.50 a, + 1.5p cos 0.50 a, C/m². Show the divergence theorem is true for…

A:

Q: (1.) Calculate the proper distance between two free particles, one at the origin of a coor- dinate…

A:

Q: 1) Consider the solid Q in the first octant, generat surfaces: z = (y – 3)² + 1, 2r + 4z = 16, 1= 1,…

A: Volume of solid: Let z=f(x, y) and z=g(x, y) be the functions of x and y of a solid, and let x=a to…

Q: 1 Evaluate dV, where E lies between the spheres x2 + y2 + z2 = 9 and x2 + y? + z2 x2 + y? + z2 36 in…

A: we want to find the triple integral ∫∫∫E1x2+y2+z2  dVwhere E lies between the sphere x2+y2+z2=9  and…

Q: Given the field D = 6p sin 124 ap +1.5p cos 12 a, C/m2, evaluate both sides of the divergence…

A: field D=6ρsin12ϕ ap + 1.5ρcos12ϕ aϕ C/m2 here ∇ = aρ ddρ + aϕddϕ+azddz Divergence = ∇·D ∇·D=…

Q: 1.Let 3 le the solid obtained by rotating by region shown in the fiqure about the Assume as3 and b=…

A: Given: y=ax(x-b)2 and y=0 and  0≤x≤b The rotation about the y-axis. Assume a=3 and b=1

Q: 5) S, is calculated by the thorem of divergence of the full range straight flux F(x, y,z) = 2xi + yj…

A:

Q: 12/ Vse the divergence of the fleld FCX4,22= a (e,3yt sin liz),9zt²+]y through the Surfacé theorem…

A: To Determine: use the divergence theorem to evaluate the flux of the field…

Q: 1. Compute the average height of the cone z = a> 0. r² + y² above the disk x² + y² ≤ a² for

A:

Q: Evaluate the triple integral III xy dV where E is the solid tetrahedon with vertices (0, 0, 0),…

A: The given integral is, ∫∫∫E xydV Here E is the solid tetrahedron with vertices O0,0,0, A10,0,0,…

Q: 1. Draw a sketch of the cylinder z= ==V36 – 4y2.

A: Given: z=1336-4y2

Q: Example 6: Evaluate ez dz dy dx E Where E is enclosed by the paraboloid Z = 3+ x? + y² , the…

A: The given problem is to evaluate the given triple integral of given function ez for which region E…

Q: 2. Suppose that the edge lengths r, y, and z of a closed rectangular box are changing at the…

A:

Q: Suppose that the edge lengths x, y, and z of a closed rectangular box are changing at the following…

A: To determine the rate at which the box’s (a) Volume, (b) Surface area and (c) diagonal length are…

Q: 12/ Vse the divergence theorem to evalvate the flux ef the fleld FC,22= a (3yt sin l2),9z+?+9y²…

A:

Q: 1 9. Evaluate dV;where E is the solid between the spheres of radius 1 (x² + y² + z²)³/2 E and 2…

A:

Q: Consider the solid Q shown in the following figure 2.x + 3y + 3z = 6 S y? + x = 2 2 If S: 2x + 3y +…

A: The given surface is as follows: 2x+3y+3z=63z=6-2x-3yz=136-2x-3yz=2-23x-y The projection of the…

Q: 12. Find the outward flux of F = x*yi-cyj+zk across the boundary of the region which is bounded…

A: