DHS 15.2 - Option 21. 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.

1.21. a (M) = (x + y - z) i - 2yj + (x + 2z) k, (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)

2.21. u (M) = x2 (y + z2), M0 (3, 0, 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)

3.21. a (M) = (y - z) i - z2j + xyzk, M0 (3, 0, 1)

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

4.21. a (M) = 6x2i + 3cos (3x + 2z) j + cos (3y + 2z) k