Use Gauss's Law to find the charge contained in the solid hemisphere  x2 + y2 + z2 ≤ a2, z ≥ 0,  if the electric field is  E(x, y, z) = x i + y j + 6z k.

Q: For F(x,y,z) = (6y-4xz/(z2+1)2)i + (-2yz-6x/(z2+1)2)j + (-3/z2+1)k,  Calculate the net flux through…

A: consider the equation

Q: Calculate the double integral || x + y)?e -r dx dy where R is the square with vertices (4, 0), (0,…

A: Given: ∫∫Rx+y2ex2-y2 dx dy where R is the square with vertices 4,0, 0,4, -4,0, 0,-4

Q: Exercise 10.6 Evaluate the line integral using a method of your choice. e¯*(cos y dx + sin y dy),…

A:

Q: Evaluate the line integral Jc F · dr, where F(x, y, z) = xi – yj + 3zk and C is given by the vector…

A: Given F(x,y,z)=xi-yj+3zk and C is the curve given by  rt=sint,cost,t. Find ∫CF·dr.

Q: Use spherical coordinates to evaluate the triple integral x² + y² + z² dV, where E is the ball: x² +…

A:

Q: Consider a thin plate of length L and width W governed by the heat equation Uf = k(uxx + uyy) for u…

A: Let  U(x,y,t)=X(x) Y(y) T(t)ut=k(uxx+uyy)T'kT=X''X+Y''Y     ......(1) Here T is function of t. and…

Q: Let S be the cylinder x +y = a°, 0szsh, together with its top, x² +y° sa, z=h. Let F= - 4yi + 4xj +…

A:

Q: x² +y? + z?e¯(2²+g² +z²), x? + y? + z2 = b² where 0 < a < b. Integrate V 'over the solid bounded by…

A:

Q: Consider the pendulum x' Y, y' = -vy - sinx, - where 0 is a friction parameter and (x, y) E S¹ x R…

A:

Q: 55. Find the flux integral, using = xi +yj + z k . (a) /7 .dÁ ,where S, is the disk x = 5, y? + z² s…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Calculate the line integral of h (x, y)= (2n) i+(x - 2y) j; over r (1) = cos u i+ sin u j, u e…

A:

Q: Find the differential of w = x ev sin z at the point (x, y, z) = (1, 1, 0). %3D O dy + dz O dy O dx…

A: Obtain the differential of the given function as follows. w=xeysin zdw=ddxxeysin zdx+ddyxeysin…

Q: 3. Find the displacement u(r, t) in a vibrating circular elastic membrane of radius 1 that satisfies…

A:

Q: Calculate the line integral of h (x, y) = (2xy) i+ (x² – 2ry) j; over r (1u1) = cos u i+ sin u j, u…

A: Here,hx,y=2xy i + x2-2xy jr(u)=cosu i+sinu j

Q: Evaluate the triple integral I/| (x, y, z) dV over the solid E. f(x, y, z) = vx2 + y², E = {(x, y,…

A:

Q: Use spherical coordinates.Evaluate the triple integral of (x2 + y2) dV,  where E lies between the…

A:

Q: Let v = zk be the velocity field (in meters per second) of a fluid in R3. Calculate the flow rate…

A: Concept: Similar to circle sphere is a two dimensional space where the set of points that are at the…

Q: Use LT and one other method to find the PS of (x' = y + z + et {y' = z + x + e* (z' = x + y + et…

A: The given system is                                   x'=y+z+ety'=x+z+etz'=x+y+et…

Q: Calculate the line integral of h (x, y) = (2xy) i + (x² – 2xy) j; over r (1) = cos u i + sin u j, u…

A:

Q: Use V(P) = k / Kx, y, z2) ds Dp(x, y, z) where k = 8.99 x 10° N - m²/C°, to compute the electric…

A: Consider the provided question, VP=k∫C∂x,y,zDpx,y,zdswhere k = 8.99 x 109 N·m2c2. To compute the…

Q: Evaluate (l. yz dV where E lies above the plane z = 0 and below the plane z = y, and inside the…

A: Given, ∫∫∫EyzdV, Where E lies above the plane z=0 and below the plane z=y and inside the cylinder…

Q: # 4 Assume that the solid has constant density k. Find the moment of inertia about the z-axis of the…

A:

Q: Given that D = zp coso a̟ C/m², calculate the charge density at (1, /4, 3) and the total charge…

A: Given that D=zρcos2ϕaz Charge density = ρv=∇.D where ∇=aρ∂∂ρ+aϕ1ρ∂∂ϕ+az∂∂z Now,…

Q: Let v = zk be the velocity field (in meters per second) of a fluid in R3. Calculate the flow rate…

A: volume ( V)= 2/3pi r3 radius of sphere in vector form can be written as = x i^ + y j^ + z k^ so…

Q: ¹ // x² + y² + 2² dV, where E is the ball: x² + y² + z² ≤ 16. Use spherical coordinates to evaluate…

A:

Q: Use cylindrical coordinates to evaluate the triple integral ∫∫∫E x^2 dV, where E is the solid that…

A: Given, E is the solid that lies within cylinder, above plane, and below the cone is

Q: (b) Let E = {(x, y, z) : –1< # < 0, 0 < y < VI– a², 0< z< 2}. Write the integral II| cos (2² + y°)dV…

A:

Q: Let E be the solid below z = 32 – a? – y² and above the square [– 4, 4] x [ – 4, 4] - Given the…

A:

Q: 1. Evaluate the integral: SSf, xe**+y<+zt dV where B is the portion of the unit ball x² + y² + z² <…

A: Here, the region B is the portion of the unit ball x2+y2+z2≤1 that lies in the first octant.…

Q: A uniform electric field points in the :-direction, E = Ek. By explicitly calculating the surface…

A:

Q: The value of the integral oIm(iz)dz, where C:| z|= 2, counterclockwise, is

A:

Q: Use 6(x, y, z) ds Dp(x, y, z) V(P) = k where k = 8.99 × 10° N m²/C2, to compute the electric…

A:

Q: A fluid motion is given by V=(y+z)i+(z+x)j+(x+y)k. Is this motion irrotational? If the velocity…

A:

Q: Verify Stokes's Theorem by evaluating F(x, y, z) = xyzi + yj + zk S: 8x + 8y + z = 16, first octant…

A:

Q: Use spherical coordinates to evaluate the triple integral e-(x²+y²+z²) x² + y² + z² where E is the…

A:

Q: Verify Stokes' Theorem for F(x, y, z) = xi + yj + y²zk where S is the first octant portion of the…

A:

Q: 1. Calculate the integral |- z+i dz throughout the contours C: z' + 2z? a.) |z +2-i|=2; b.) |z – 21|…

A:

Q: d the mass of a wire in the shape of helix r(t) = cos лti+sin лtj+8tk, ≤t≤1, where the density of…

A:

Q: Use cylindrical coordinates to evaluate the triple integral /// (æ – y) dV, where E is the solid…

A:

Q: aluate the line integral fc F dr, where F(x, y, z) =3 sin xi + 2 cos yj+ 4xzk and C is given by the…

A:

Q: Find the work done in moving a point charge Q- - 20 1C from.the origin to (4,0,0) m in the ficld E-+…

A: To find - Find the work done in moving a point charge Q = -20 μC from the origin to 4, 0, 0m in the…

Q: Calculate the line integral of h (x, y) = (2xy) i + (x² – 2xy) j; over r (u) = cos u i+ sin u j, u e…

A:

Q: Given that D=p²z?sina,C/m², calculate the total charge enclosed by a cylinder with radius = R1 meter…

A: Total charge enclosed = volume integral (volume charge density dv) volume charge density =…

Q: For an infinitely long conducting cylinder of radius a, with its axis coincident with its z-axis,…

A: The voltage is u(r,θ) It satisfies the Laplace equation, ∇2u=0 --&gt;equation(1) Equation(1)…

Q: Evaluate the line integral INTEGRAL C (x − y 2 )ds where C is given by r(t) = (t^2 + 1)i + (t +…

A:

Q: Find the moment of mass from the x-axis for y = x In(x) – 3x and y density is equal to 0.37…

A:

Q: The electric potential at each point (x,y) in the region R: x 2 0, y s1 is given by v - 48xy – 32x -…

A:

Q: (7) Determine the upward flux of V x F through the part of the elliptical cone C = {(x, y, z)2 =…

A:

Q: tes to evaluate the triple integral e Vz² + y² + pounded by the spheres z² + y² + z² = 4 and z? + y…

A: Given, ∫∫∫Ee-x2+y2+z2x2+y2+z2dV E is the region bounded by spheres x2+y2+z2=4 and x2+y2+z2=25.

Q: Calculate the double integral || (x + y)?e* - Ý dx dy where R is the square with vertices (2, 0),…

A: According to the given information, it is required to find the integral over R where R is square.