DHS 15.1 - Option 14. 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)

1.14. u (M) = ln (1 + x + y2 + z2), M1 (1, 1, 1), M2 (3, -5, 1)

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): 2x + y + z = 4

3. Calculate the surface integral of the second kind.

 where S - the surface of the hyperboloid x2 + y2 = z2 + 1 (the normal vector n which forms an obtuse angle with the unit vector k), cut off by the plane z = 0 and z = √3.

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.

4.14. and (M) = 4xi + (x - y - z) j + (3y + 2z) k, (p): 2x + y + z = 4