DHS 15.1 - Option 12. 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)

1.12. u (M) = y2z - 2xyz + z2, M1 (3, 1, -1), M2 (-2, 1, 4)

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): 3x + 2y + 2z = 6

3. Calculate the surface integral of the second kind.

 where S - the surface of the cone z2 = x2 + y2 (normal vector n which forms an obtuse angle with the unit vector k), which lies between the plane z = 0, z = 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.

4.12. and (M) = xi + (y - 2z) j + (2x - y + 2z) k, (p): x + 2y + 2z = 2