14. (a) Verify Stokes' theorem for F = −2yi + rzj - yz²k and S is the surface of the paraboloid 2z+y2 0≤=≤2 (b) Verify Stokes' theorem for F as in (a) and the surface being the disc r² + y² ≤4, z = 2.

Let S be the portion of the plane 2x + 3y + z = 2 lying between the points (−1, 1, 1), (2, 1, −5), (2, 3, −11), and (-1, 3, -5). Find parameterizations for both the surface S and its boundary S. Be sure that their respective orientations are compatible with Stokes' theorem. from (-1, 1, 1) to (2, 1,-5) from (2, 1, 5) to (2, 3, -11) from (2, 3, -11) to (-1, 3, -5) from (-1, 3, 5) to (-1, 1, 1) boundary S₁ (t) = S₂(t) = S3(t) = S4(t) = Φ(u, v) = te [0, 1) te [1, 2) te [2, 3) te [3, 4) UE [-1, 2], VE [1, 3]

Use Stokes' Theorem to evaluate of intersection of the plane x + 3y +z = 12 with the coordinate planes. (Assume that C is oriented counterclockwise as viewed from above.) F. dr where F = (x + 6z)i + (8x + y)j + (10y −z) k_and C is the curve

Let S be the portion of the plane 2x + 3y + z = 3 lying between the points (-1, 1, 2), (2, 1, -4), (2, 3, -10), and (-1, 3, -4). Find parameterizations for both the surface S and its boundary as. Be sure that their respective orientations are compatible with Stokes' theorem. from (-1, 1, 2) to (2, 1, -4) S,(t) tE [0, 1) from (2, 1, -4) to (2, 3, -10) S2(t) tE[1, 2) from (2, 3, -10) to (-1, 3, -4) S,(t) = tE [2, 3) from (-1, 3, -4) to (-1, 1, 2) s,(t) tE [3, 4) boundary Ф(и, v) uE [-1, 2], v E [1, 3] II

Find a parametrization of the curve of intersection of the surfaces z = x² – y² and z = x2 + xy + 1.

4. Consider the surface defined by the portion of the cone z = -Vx2 + y? that lies below the rectangle (x, y) E [–1, 1] x [2, 4]. (a) Find parametric equations for this surface. (b) Compute the area of this surface.

4. Find parametric equations of the line tangent to the surface z = y² + x³y at the point (2,1,9) whose projection on the xy-plane is parallel to the x-axix.

Find the distance between the point (0,0,0) and the plane 2x – y + 2z = 4.

شكتذمسخaobckqid

If a parametric surface given by r₁(u, v) = f(u, v)i + g(u, v)j + h(u, v)k and -4 ≤ u ≤ 4,-2 ≤ v ≤ 2, has surface area equal to 8, what is the surface area of the parametric surface given by r2(u, v) = 5r₁(u, v) with −4 ≤ u ≤ 4, −2 ≤ v ≤ 2? 27