The flux the vector field through the specified surface approximately

Solved Step by Step With Explanation-Flux through a surface

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Solved Step by Step with Explanation- Flux through a surface

3.32.Answer

ρ = 2 (where ρ is the radial distance from the origin)

0 < ϕ < π/2 (polar angle)

Step 1: Find the unit normal vector (n̂)

The unit normal vector (n̂) to the surface can be determined using the gradient of the scalar function ρ, which is simply the radial distance from the origin.

n̂ = (∇ρ) / |∇ρ|

= (2ax) / |2ax|

Where dS is the differential area vector of the surface.

In this case, the flux integral simplifies to:

Φ = ∬S A · n̂ dS

= ∫[0 to 0.1] ∫[0 to π/2] (2xax - z^2ay + 3xyaz) · ax (ρ^2 sin(ϕ) dϕ dz)

Φ = ∫[0 to 0.1] ∫[0 to π/2] (2ρ sin(ϕ)) (ρ^2 sin(ϕ) dϕ dz)

= 2 ∫[0 to 0.1] ∫[0 to π/2] ρ^3 sin^2(ϕ) dϕ dz

= π/2 ∫[0 to 0.1] ρ^3 dz

= π/2 * (1/4) * (0.1)^4