Let Surface S be that portion of the cylinder x2 + y2 = 9, which lies between the planes z = y and z = 6. a.) Sketch the Surface S. b.) Parametrize the Surface S. c.) Evaluate the following Surface Integral: ∫∫(y-z)dS
5. Calculate the surface area of the portion of the sphere x2+y2+12-4 between the planes z-1 and z ะไ 6. Evaluate (xyz) dS, where S is the portion of the plane 2x+2y+z-2 that lies in the first octant. 7. Evaluate F. ds. a) F = yli + xzj-k through the cone z = VF+ア0s z 4 with normal pointing away from the z-axis. b) F-yi+xj+ek where S is the portion of the cylinder+y9, 0szs3, 0s r and O s y...
(20) Let S be the sphere x2 + y2 + x2 = 4 with outward normal vector. Let F(x, y, z) = (2-3 + tan-'(yz), ex2+2 + y3, cosh(xy) + 23). Use the Divergence Theorem to find the flux of Ě out of S.
Q1. Evaluate the line integral f (x2 + y2)dx + 2xydy by two methods a) directly, b) using Green's Theorem, where C consists of the arc of the parabola y = x2 from (0,0) to (2,4) and the line segments from (2,4) to (0,4) and from (0,4) to (0,0). [Answer: 0] Q2. Use Green's Theorem to evaluate the line integral $. F. dr or the work done by the force field F(x, y) = (3y - 4x)i +(4x - y)j...
Let E be the solid that lies inside the cylinder x^2 + y^2 = 1,
above the xy-plane, and below the plane z = 1 + x. Let S be the
surface that encloses E. Note that S consists of three sides: S1 is
given by the cylinder x^2 + y^2 = 1, the bottom S2 is the disk x^2
+ y^2 ≤ 1 in the plane z = 0, and the top S3 is part of the plane z...
Verify Stokes, Theorem for the surface S that is the paraboloid given by z = 6-x2-y2 that lies above the plane z 2 (oriented upward) and the vector field F(x, y, z)2yzi+yj+3xk.
Verify Stokes, Theorem for the surface S that is the paraboloid given by z = 6-x2-y2 that lies above the plane z 2 (oriented upward) and the vector field F(x, y, z)2yzi+yj+3xk.
= and z= 8. Let A be the part of the cylinder x2 + y2 1 between the planes z = 2, where n points away from the z-axis. Let C be the counterclockwise boundary of A. Let F(x, y, z) = (2xz + 2yz, –2xz, x2 + y²). Verify Stokes' Theorem: (a) Evaluate the line integral in Stokes' Theorem. (Hint: C has two separate parts.] (b) Evaluate the surface integral in Stokes' Theorem. Hint: curl (F) = (2x +...
Let F(r, y, z)(z4+ 5y3)i + (y2 surface of the solid octant of the sphere x2+yj2 + 22 = 9 for x> 0, y> 0 and z> 0 )j+ (3z + 7)k be the velocity field of a fluid. Let B be the Determine the flux of F through B in the direction of the outward unit normal
Let F(r, y, z)(z4+ 5y3)i + (y2 surface of the solid octant of the sphere x2+yj2 + 22 = 9 for x>...
3. (5 points) Use the Divergence Theorem to find the outward flux of the vector field F(x, y, z) - 3ry? i + xe'j + 23k across the surface of the solid bounded by the cylinder y2 + z-1 and the planes z =-1 and x = 2.
3. (5 points) Use the Divergence Theorem to find the outward flux of the vector field F(x, y, z) - 3ry? i + xe'j + 23k across the surface of the solid...
(i) The sides of a given grain silo are represented by the equation of the cylinder x2 +y-3. The top of the silo is the portion of the sphere x2 + y2 + z2-7 lying within the cylinder and above the zy plane. Sketch and find the volume of the silo using an appropriate coordinate system Q2. [10] (ii) Given that C is the boundary of the plane 2x +2y+z = 6 that lies in the first octant and F...