Solution of IDZ Ryabushko 15.2 - Variant 2

Section: Chapter 15. Elements of Field Theory
Topic: Classification of vector fields
Description: Determining the type of vector field: potential (irrotational), solenoidal (tubular), harmonic. Finding the field potential. Checking the conditions for potentiality, solenoidal nature, and harmonicity.

Tasks:
1. Compute the circulation of the vector field a(M) around the contour of the triangle obtained by intersecting the plane (p): Ax + By + Cz = D with the coordinate planes, with positive direction of traversal relative to the normal vector n = (A, B, C) of this plane, in two ways: 1) using the definition of circulation; 2) using Stokes' formula.
2. Find the magnitude and direction of the greatest change of the function u(M) = u(x, y, z) at the point M₀(x₀, y₀, z₀).
3. Find the greatest circulation density of the vector field a(M) at the point M₀(x₀, y₀, z₀).
4. Determine whether the vector field a(M) is potential, solenoidal, or harmonic.

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