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Analytical Evaluation of Line and Surface Integrals in Multidimensional Spaces

In the realm of advanced calculus and geometric analysis, the computation of line and surface integrals stands as a fundamental aspect, crucial for understanding the intricate properties of fields and surfaces in multidimensional spaces. This discourse aims to meticulously examine and compute the surface integral of a quadratic function over a spherical domain and the line integral of a polynomial function across the boundary of a square region. Through a systematic approach, we will parameterize the necessary curves, adopt appropriate coordinate transformations, and implement precise mathematical integration techniques. This exploration not only illuminates the theoretical aspects of calculus but also exemplifies its practical significance in analyzing and interpreting the spatial and dynamical attributes of complex geometrical entities.
POSTED BY: Athanasios Paraskevopoulos
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Can we calculate example 3 with LineIntegrate ?

I tried: 

  LineIntegrate[y^2 - x, {x, y} \[Element] Line[{{-1, -1}, {-1, 1}, {1, 1}, {1, -1}, {-1, -1}}]]
 (* 16/3 *)

and it shouldn't be zero ?

Regards M.I.
POSTED BY: Mariusz Iwaniuk

Yes it should be zero

POSTED BY: Athanasios Paraskevopoulos 
LineIntegrate[{x^2, -y}, {x, y} \[Element] Line[{{-1, -1}, {-1, 1}, {1, 1}, {1, -1}, {-1, -1}}]]
LineIntegrate[{y^2, -x}, {x, y} \[Element]  Line[{{-1, -1}, {-1, 1}, {1, 1}, {1, -1}, {-1, -1}}]]

(* 0 *)
(* 4 *)

I tried with Maple 2024 gave me the same answers.
POSTED BY: Mariusz Iwaniuk

Oh thank you very much. I will fix it
POSTED BY: Athanasios Paraskevopoulos

Isn't there a mistake on that particular code?
POSTED BY: Athanasios Paraskevopoulos
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