DHS 15.2 - Option 8. 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.

1.8. a (M) = (x + z) i + zj + (2x - y) k, (p): 2x + 2y + z = 4

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)

2.8. u (M) = y2z - x2, M0 (0, 1, 1)

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

3.8. a (M) = xzi - xyzj + x2zk, M0 (0, 1, 1)

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

4.8. a (M) = yzi + (x - y) j + z2k