DHS 15.2 - Option 27. 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.

1.27. a (M) = (x + z) i + (z - x) j + (x + 2y + z) k, (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)

2.27. u (M) = x2y - z, M0 (-2, 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)

3.27. a (M) = (x - z) i + xyzj + xk, M0 (-2, 2, 1)

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

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