DHS 15.2 - Option 25. 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.

1.25. a (M) = (x + z) i + 2yj + (x + y - z) 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)

2.25. u (M) = x2z - y2, M0 (1, 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)

3.25. a (M) = xzi + (x - y) j + x2zk, M0 (1, 1, -2)

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

4.25. a (M) = 3x2i + 4 (x - y) j + (x - z) k