„Consider the space (R,t). .where t={R,Ø}U{ACR:[0,1)CA} f A=[0,1], then the interior of A is (0,1) O [0,1] O (0,1] O

Q: „Consider the space (R,T). where t={R,Ø}U{ACR:[0,1)CA} :Then (R,T) is not T, O T, but not T, O T2 O…

A:

Q: y dV W

A: Let us consider the triple integral ∫ab∫h(x)g(x)∫p(x,y)q(x,y)f(x,y,z)dzdydx To solve the integral…

Q: Use Gauss' theorem to evaluate F.ndS for the given F and o. (c) F(x, y, z) = x² i+ y²j+ z² k, o :…

A: Let's find.

Q: Let (r, y, z) 2 F := + y2 + and let S be the boundary of the sphere x2 + y² + z2 = 1 oriented…

A: Divergence theorem: Let D be a simple solid region and s is the boundary surface of D. Let F→ be a…

Q: Let F=[-x, y, z] and S be a sphere of radius R. Using the divergence theorem, the value of the…

A: For evaluation of the given integral, we must know the direction of normal to the surface S. For…

Q: Evaluate both sides of divergence theorem for A = pa,+pcos(@)a , + za , through the closed surface…

A: Given vector field is  Divergence theorem

Q: Evaluate : //| cos(x² +y³) dV, where E is the region bounded by the disk x² + y? < 9 E and the plane…

A: From the region of integration which is bounded by x2+y2≤9 and z=1, identify the limits of x, y and…

Q: Let E be the region bounded by the hemisphere x² + y? + (z – 2)² = 4, 2 < z < 4, and the plane %3D z…

A:

Q: Given F(x, y, z) = -yzi + xzj – xyzk be a vector function that passes through a surface S of z + x2…

A:

Q: Let F=[-x², 3y, 2xz] and S be a sphere of radius R. Using the divergence theorem, the value of the…

A: The answer is option D. Divergence theorem  Let E be a simple solid region and S is the boundary…

Q: Use the Divergence Theorem to evaluate F. ds, where F(x, y, z) = z?xi + (+ tan zj + (x²z + y2)k and…

A: We are given a closed surface integral and we have to use the Divergence Theorem to evaluate that.

Q: Consider the surface S: z = 4 - x2 ,bounded by y = −5(x2 - 1) and y = 0, just as it is shown in the…

A: We have to find the area of the surface S, and we will use surface integral to do that.

Q: Compute the triple integral f(x,v.z)dxdydz where D is the sphere with center O and radius R and…

A: Given that ∭Df(x,y,z)dxdydz where f(x,y,z)=R2-(x2+y2+z2). By the spherical coordinates,…

Q: Identify the theorem: If S is the boundary surface of a solid E in 3-dimensional space, then SS, F.…

A:

Q: Let C be the positively oriented square with vertices 0, 1, 1 + i, i. Evaluate using the definition…

A:

Q: Let F= [x, y, z] and S be a sphere of radius R. Using the divergence theorem, the value of the…

A: Given : F = x,y,z

Q: Verify the divergence theorem, ffaw F · dS = S[fw, Div FdV, for F(x, y, z) = (z)i + (y)j + (x)k,…

A:

Q: (9) Evaluate the surface integral / S, czdS, where o is the part of the plane r+2y+3z = 6 that lies…

A:

Q: Let d: Rx R*→[0, ∞) such that d(XF)=√(x, y)² show that (R.d) metric space.

A: `

Q: Consider the following maps from the closed unit disk D = {(x, y) E R? : x? + y < 1} in R? to…

A:

Q: If F = xy?i + yz? j + zx2k, and S is the region bounded by the triangle with vertices (1,0,0),…

A:

Q: Let F(z, y, 2) = 12?ri + (ty + tan(z))j + (1z?z+ 4y?)k. Use the Divergence Theorem to evaluate f, F…

A: We must apply the Divergence theorem on a closed solid, but the surface S is not closed (it is just…

Q: Calculate F. dS where F = (e" sin y, e* cos y, yz“) S is the surface of the box bounded by the…

A:

Q: Use Stoke's theorem to compute the integral SL(V × F) · n dS, where F(x, y, z) = (xz, yz, xy) and S…

A: We have to compute ∬(∇×F)·n dS , where F(x,y,z)=(xz,yz,xy) and S is the portion of the sphere…

Q: 2. Use Stokes' Theorem to evaluate , Vx F ·n dS where F(x, y, z) = and C is the part of the sphere…

A:

Q: Verify the Divergence Theorem SI div F av F. dS S E for F(r,y,2) = z i+ yj+xk and where E is the…

A: Given that Fx,y,z=z i+ y j +x k and E is the solid ball x2+y2+z2≤16. First find the divergence of F.…

Q: Let F=[-x, y, z] and S be a sphere of radius R. Using the divergence theorem, the value of the…

A: Hi, since you have asked multiple questions, we will solve the only one question for you. If want…

Q: 5. Compute / | F. aš, where F = (y, -x, 2z) and S is the surface of the sphere of radius 2, centered…

A:

Q: Let F = [-x, 3y, 2xz] and S be a sphere of radius R.Using the divergence theorem, the value of the…

A:

Q: The flux of F(x, y, z) = 2³î – r³}+ y®k, where o is the sphere | x2 + y? + z2 = a² with outward…

A:

Q: Use Gauss’ theorem to evaluate F.ndS for the given F and o. (b) F(x, y, z) = x³ i+x?yj+xy k, o : the…

A: We will use gauss divergence theorem to find value of given surface integral  If E is a  simple…

Q: (a) Let S be the surface z2 = x² + y? bounded between the planes z = 0 and z = 2. Evaluate (z –…

A:

Q: Compute the contour integral dz, where l'is the contour of the perimeter of the square with vertices…

A: We will do it  by dividing r into 4 paths.

Q: Prove that dr-V3Ta? where こis the intersection of the plates: x²+ and and F=ly,Z,x)

A:

Q: Find the volume of the Intersection curve C C Projection of C on ay-plane

A: Given: Given paraboloids are z=x2+y2 and z=8-x2-y2 We want to find the volume bounded by the…

Q: Let F(x, y, z) = 2z? xi + (y + tan(z))j + (2x²z – 2y²)k. Use the Divergence Theorem to evaluate F.…

A:

Q: Show that the mapping w = (1 – j)z, where w = u + jv and z = x + jy, maps the region y > 1 in the z…

A:

Q: Use Gauss' theorem to evaluate F.ndS for the given F and o. (b) F(x, y,z) = x³i+x²yj+xyk, o : the…

A: Gauss theorem states that ∫∫σ F.ndS=∬∫V div(F) dVwhere V is the volume inclosed by the closed…

Q: Let E denote the part of the plane x + y + z = 1 that is inside the cylinder r? + y? = 1. Compute…

A: We have to find the value of  ∬ F→.dS→

Q: Let R be the three-dimensional region that lies below z= xy and above the triangle in the ry palne…

A: Please see the explanation below

Q: Let A = 6 zi + (2x + y)j – x k ; Evaluate A.ndS over the entire surface S of the region bounded by…

A: Please see the figure below to comprehend what's going on:

Q: Use the Divergence Theorem to evaluate F. dS where S is the sphere r² + y² + z² = 9 and F = (3r, 2y,…

A:

Q: „Consider the space (R,t). .where t={R,Ø}U{ACR:[0,1)CA} :lf A=(0,1), then the closure of A is (0,1)…

A:

Q: Evaluate both sides of divergence theorem for A = pa,+ pcos(o)a,+ za, through the closed surface…

A: Given A=ρaρ+ρcosϕaϕ+zaz and the closed surface is 0≤ρ≤1, 0≤ϕ≤π, 0≤z≤2. We have to evaluate the both…

Q: Let S be the piece of the surface z = 9-r that lies inside the cylinder r2 +y² = 9, and let F = (x,…

A: Given, S be the piece of the surface z=9-x2 that lies inside the cylinder x2+y2=9, and let F=x,y2,z.…

Q: If S is an arbitrary closed surface covering a V and A = axi + byj + czk, then www prove that f. A.n…

A:

Q: 1. Evaluate the surface integral SL xz² ds ; o is the part of the cone z = Vx2 + y² that lies…

A:

Q: Let S be the upper surface of the sphere x- 2x+y+z = 0 and let r be the boundary. Find [.curi(F) -…

A: S is upper surface, the equation  x2-2x+y2+z2=0  x-12+y2+z2=1  Hence, is a sphere centered at (1, 0,…

Q: Let F(x, y, 2) = 32?ai+ (y³ + tan(z))j+ (3x²z- 4y?)k. Use the Divergence Theorem to evaluate fs F dS…

A:

Q: Verify the Divergence Theorem for the F(x,y,z) = 4xi +4yj +2z? k over the surface S of x² +y² and z…

A: