Use Gauss' theorem to evaluate F.ndS for the given F and o. (a) F(x, y, z) = (x – z) i+ (y – x)j+(z – y) k, o : the closed surface enclosing the solid bounded by the cylinder x² + y² = 4 and the planes z – 0 and z

Q: c) The surface S is the portion of the plane x+y+z=1 in the first quadrant of the Oxyz Cartesian…

A: Given: The surface and curve is shown below   The region s is the triangle and the curve C is the…

Q: Let (S) be the part of the sphere (x – 3)² + y² + (z – 1)² = 4 tha Then (S) is the graph of the…

A:

Q: Evaluate ff, (F n) dS in the following cases: (a) F = (2y+z, x+z, y − x) and S is the portion of the…

A: (a)

Q: Find a number a so that the change of coordinates s = x + ay, t = y transforms the integral ff dady…

A:

Q: Use Stokes' Theorem to evaluate F dr where F = (x+ 9z) i + (5x + y) j + (7y- z) k and C is the curve…

A:

Q: F · dr  where F  =  (x + 5z) i  +  (7x + y) j  +  (2y − z) k   and C is the curve of intersection of…

A:

Q: Verify Green's Theorem in the plane for F = xyî + x²ĵ and where C is the boundary of he region…

A:

Q: Use Stokes Theorem to evaluate •[F F-dr where F(z, y, z)=zi+si+z²+ y²)k and C is the boundary of the…

A: The given vector field is F=xi+yj+9x2+y2k. Our aim is to find the value of line integral using the…

Q: Find the circulation f V. dR, where V = yi + (x² + y²)j + (x + y)k, and C' is the triangle with…

A: In this question we will find the circulation of

Q: The volume of the solid bounded by the plane 3x+6y-2z=12 and the coordinate planes is O A. 2-2y+4 1…

A: We need to find the volume of the solid bounded by the plane 3x+6y-2z=12. Now,…

Q: %3D by the paraboloid z 4-x2-12 =4-x- c.m. R. x +y = 4

A: Given, Solid of constant density δ bounded by R: x2+y2≤4 in plane z=0 and paraboloid z=4-x2-y2 To…

Q: F = (x² – y²,x, 2yz); C is the boundary of the plane z = 6 – 2x – y in the first octant.

A: Given, F=x2-y2,x,2yz Find the line integral ∮CF·dr in the following. Here C is the boundary of the…

Q: a) Evaluate ff, (2x – z) dS, where S is part of the plane 3x + 2y +z = 1 that lies above the…

A:

Q: Find the maximum value of F. dr. where F= 2xy²1+ (22-6xy²)j + (4y - 2x2y) k and C is a simple closed…

A: Given: F→=2xy2i+2z-6xy2j+4y-2x2yk and c is a simple closed curve in the plane 3x+y+z=6 oriented…

Q: Use Stokes' Theorem to evaluate fF.dr where F(x, y, z) = zi − 3yzj+ xk and C is the boundary of the…

A:

Q: A scalar function F(x,y,z) = xy2 + x2z – z2y. Find the gradF at (1,1,1).

A: A scalar function is given. We have to find the gradient at a specified point.

Q: · ƒ [‚²³ · F . S F = (4x³z, 4y³z, 3z¹) Calculate dS where and S is the surface of the solid bounded…

A:

Q: a) Evaluate [-ri-yj+2: k) - n ds, (-a i- where a is the portion of the paraboloid z = 3r + 3y…

A:

Q: Verify Stoke's theorem for A=(2x-y)i-yz²j-y*zk where S is the upper half 5A. surface of the sphere *…

A: Consider the given vector and sphere equation. A→=2x−yi−yz2j−y2zk and x2+y2+z2=1 Since S is the…

Q: Evaluate x2 zdS, where S is the part of the S plane z = 1+ 2x + 3y above the rectangle [0, 2] × [0,…

A: We have to evaluate ∬Sx2zdS, where S is the part of the plane z=1+2x+3y above the rectangle 0,2×0,3.…

Q: b) Use Gauss' theorem to evaluate RR F. ndS where F = (x² - y)i + y³z²j + (xy – y²z³)k UTM UTM and o…

A:

Q: a) Evaluate ff, (2x – z) dS, where S is part of the plane 3x + 2y + z = 1 that lies above the…

A: Surface integral

Q: Let F = (z, 0, y) and let S be the oriented surface parametrized by Þ(u, v) = (u² – v, u, v²) for 0…

A: From the given information:

Q: 1 E x² + y² + z² 2 = 25 and x² + y² + z² Evaluate J x² + y² + z² dV, where E lies between the…

A: Given that the spheres,          x2+y2+z2=25, and x2+y2+z2=49, in the first octant.

Q: A. Let f(x, y, z) = z²-x² - y² .The set of all values of c such that the level surface f(x, y, z) =…

A: