DHS 15.2 - Option 4. Decisions Ryabushko AP

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• 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.

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

• 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)

2.4. u (M) = xyz2, M0 (3, 0, 1)

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

3.4. a (M) = xzi + zj + yzk, M0 (3, 0, 1)

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

4.4. a (M) = (x2 - z2) i - 3xyj + (y2 + z2) k

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