DHS 15.2 - Option 5. 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.

1.5. a (M) = (y + z) i + (x + 6y) j + yk, (p): x + 2y + 2z = 2

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)

2.5. u (M) = x2y2z, M0 (-1, 0, 3)

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

3.5. a (M) = xyi + xyzj - xk, M0 (-1, 0, 3)

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

4.5. a (M) = 2xyzi - y (yz + 1) j + zk