DHS 15.2 - Option 2. Decisions Ryabushko AP

1. Calculate and circulation of the vector field (M) over the contour of a triangle obtained by intersection of the plane (p): Ax + By + Cz = D with the coordinate planes, with respect to the positive direction of the normal vector bypass n = (A, B, C) this plane in two ways: 1) using the definition of circulation; 2) using the Stokes formula.

1.2. a (M) = (x + z) i + zj + (2x - y) k, (p): 3x + 2y + z = 6

2. Find the magnitude and direction of the greatest changes in the function u (M) = u (x, y, z) at the point M0 (x0, y0, z0)

2.2. u (M) = x2yz, M0 (2, 0, 2)

3. Find the greatest density of the circulation of the vector field a (M) = (x, y, z) at the point M0 (x0, y0, z0)

3.2. a (M) = xyi + yzj + xzj + xzk, M0 (2, 0, 3)

4. Determine whether the vector field a (M) = (x, y, z) solenoidal

4.2. a (M) = x2yi - 2xy2j + 2xyzk