Reshebnik Arutyunov Yu N401 Setting Higher Mathematics

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A ready solution to the problem. The sorcerer of Arutyunov. Assignment №401.

401 - 410. Given a vector field F = Xi + Yj + Zk and the plane Ax + By + Cz + D = 0 (p), which together with the coordinate planes forms a pyramid V. Let σ be the base of the pyramid belonging to the plane (p); Λ is the contour bounding σ; N is the normal to σ, directed outside the pyramid V. It is required to calculate:

• the flux of the vector field F through the surface σ in the direction of the normal n;
• the circulation of a vector field F along a closed contour λ directly and applying the Stokes theorem to the contour λ and the bounded surface σ with the normal n;
• the flux of the vector field F through the complete surface of the pyramid V in the direction of the outer normal to its surface directly and applying Ostrogradsky's theorem. Draw a drawing.

• F = (x + z) i; X + y + z = 2 = 0

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Detailed solution. Decorated in Microsoft Word 2003 (Quest decided to use the formula editor)