Consider the vector field \( \mathbf{F}(x, y, z)=\frac{(x, y, z)}{\|(x, y, z)\|^{3}} \) and the surface of the ellipsoid \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 \), where \( a, b, c \) are distinct digits of your student ID. Considering the outward orientation, calculate: 1. the flux through this surface without using Gauss' theorem; 2. the flux through this surface using Gauss' theorem. Sketch the surface and the vector field.
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The flux through the surface without using Gauss' theorem: The flux of a vector field F through a surface S is given by the surface integral of F over S. In this case, the surface S is the surface of the ellipsoid, and the vector field F is given by F(x, y, z) = Show more…
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