DHS 15.2 - Option 10. Decisions Ryabushko AP

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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.

1.10. a (M) = (2y - z) i + (x + y) j + xk, (p): x + 2y + 2z = 4

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)

2.10. u (M) = x (y + z), M0 (0, 1, 2)

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

3.10. a (M) = xzi - yj - zyk, M0 (0, 1, 2)

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

4.10. a (M) = 3x2yi - 2xy2j - 2xyzk