DHS 15.2 - Option 7. 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.

1.7. a (M) = (3x + y) i + (x + z) j + yk, (p): x + 2y + z = 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)

2.7. u (M) = xy2z2, 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)

3.7. a (M) = y2i - xyj + z2k, M0 (-2, 1, 1)

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

4.7. a (M) = (x2 - y2) i + (y2 - z2) j + (z2 - x2) k