DHS 15.2 - Option 6. 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.

1.6. a (M) = (y + z) i + (2x - z) j + (y + 3z) k, (p): 2x + y + 3z = 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)

2.6. u (M) = x2yz2, M0 (2, 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)

3.6. a (M) = yzi - z2j + xyzk, M0 (2, 1, -1)

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

4.6. a (M) = (2x - 3y) i + 2xyj - z2k