Solution of IDZ Ryabushko 15.1 - Variant 26

Section: Chapter 15. Elements of Field Theory
Topic: Vector fields: flux, divergence, circulation, curl
Description: Calculation of the gradient of a scalar field, directional derivative. Finding the flux of a vector field through a surface, divergence, circulation along a contour, curl. Application of the Ostrogradsky-Gauss and Stokes formulas.

Tasks:
1. Given a function u(M) = u(x, y, z) and points M₁, M₂. Compute: 1) the derivative of this function at the point M₁ in the direction of the vector M₁M₂; 2) grad u(M₁).
2. Compute a surface integral of the first kind over the surface S, where S is the part of the plane (p) cut off by the coordinate planes.
3. Compute a surface integral of the second kind.
4. Compute the flux of the vector field a(M) through the outer surface of the pyramid formed by the plane (p) and the coordinate planes, in two ways: a) using the definition of flux; b) using the Ostrogradsky – Gauss formula.

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