DHS 15.1 - Option 10. Decisions Ryabushko AP

1. Dana function u (M) = u (x, y, z) and the point M1, M2. Calculate: 1) The derivative of this function in the direction of the point M1 M1M2 vector; 2) grad u (M1)

1.10. u (M) = 5x2yz - xy2z + yz2, M1 (1, 1, 1), M2 (9, 3, 9)

2. Calculate the surface integral of the first kind on the surface S, where S - part of the plane (p), cut off by the coordinate planes.

 (P): x + 2y + z = 2

3. Calculate the surface integral of the second kind.

 where S - the surface of the paraboloid z = x2 + y2 (normal vector n which forms an obtuse angle with the unit vector k), cut out by the cylinder x2 + y2 = 1.

4. Calculate the flow vector field a (M) through the outer surface of the pyramid formed by plane (p) and the coordinate planes in two ways: a) determination using flow; b) using the formula Ostrogradskii - Gauss.

4.10. and (M) = (x + y - z) i - 2yj + (x + 2z) k, (p): x + 2y + z = 2