DHS 15.2 - Option 30. 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.

1.30. a (M) = 3xi + (y + z) j + (x - z) k, (p): x + 3y + z = 3

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)

2.30. u (M) = z (x + y), M0 (1, -1, 0)

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

3.30. a (M) = (x - z) i - yj + xyk, M0 (1, -1, 0)

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

4.30. a (M) = (y - z) i + (z - x) j + (x - y) k