DHS 15.2 - Option 23. 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.

1.23. a (M) = (2y + z) i + (x - y) j - 2zk, (p): x - y + 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)

2.23. u (M) = x (y2 + z2), M0 (1, -2, 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)

3.23. a (M) = z2i - xzj + z2k, M0 (1, -2, 1)

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

4.23. a (M) = 3 (x - z) i + (x2 - y2) j + 3zk