Answer :
The net outward flux across the boundary of the tetrahedron is 5, using the concept of the gradient of a function.
The gradient function in a vector field
The gradient function is related to a vector field and it is derived by using the vector operator ∇ to the scalar function f(x, y, z). The gradient is a fancy word for derivative or the rate of change of a function. It's a vector (a direction to move) that. Points in the direction of greatest increase of a function are zero at a local maximum or local minimum because there is no single direction of increase
Vector field:
F = ( -x, 3y, 2 z )
Δ . F = (i δ/δx + j δ/δy + k δ/δz) (-x, 3y, 2 z )
Δ . F = [δ/δx(-x)] + δ/δy (3y) + δ/δz (2z)]
Δ . F = - 1 + 3 + 2
Δ . F = 4
According to divergence theorem
Divergence Theorem
The divergence theorem states that the surface integral of the normal component of a vector point function F over a closed surface S is equal to the volume integral of the divergence. F took over the volume V enclosed by the surface S. The divergence theorem says that when adding up all the little bits of outward flow in a volume using a triple integral of divergence, the total outward flow from that volume, as measured by the flux through its surface.
Flux = ∫∫∫ Δ. (F) DV
x+ y +z = 1; so, 1st octant
x from 0 to 1
y from 0 to 1 -x
z from 0 to 1-x-y
∫₀¹∫₀¹⁻ˣ∫₀¹⁻ˣ⁻y (4) dz dy dx
= 4 ∫₀¹∫₀¹⁻ˣ (1 - x - y) by dx
= 5
Therefore, conclude that the net outward flux across the boundary of the tetrahedron is 5
To learn more about Gradient Function visit:
brainly.com/question/29097153
#SPJ4
