DHS 15.2 - Option 24. 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.

1.24. a (M) = (3x - y) i + (2y + z) j + (2z - x) k, (p): 2x - 3y + z = 6

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)

2.24. u (M) = x2 + 3y2 - z2, M0 (0, 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)

3.24. a (M) = xyi + (x - z) j + (y - x) k, M0 (0, 0, 1)

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

4.24. a (M) = (2x - yz) i + (xz - 2y) j + 2xyzk