Question

Why is the upward flux of a vertical vector field f0,0,1 across a surface equal to the area of the projection of the surface in the xy-plane?

Answer 1

The area of the projection of the surface in the xy-plane <-2x, -2y, -1>.

What is area of the projection?

By projecting a three-dimensional object's form onto any plane, one may estimate the two dimensions of its area. In particular, hardness testing, axial stress, wind pressures, and terminal velocity are frequently utilised in domains of mechanical engineering and architectural engineering that are connected to this.

$Flux = \int\limit\int\limits_{s} F.N . ds=\int\limits\int\limitsx_{R} < 0,0,1 > < -z_{x} ,-z_{y}, -1 > dA$

$=\int\limit\int\limitsx_{R} dA$

normal vector N = <-2x, -2y, -1>

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