a. Show that the outward flux of the position vector field 𝐅= x 𝐢+y 𝐣+z 𝐤 through a smooth close

step 1 We have our f function, which is equal to x times i hat plus y times j hat plus z times k hat. N

A. Show that the outward flux of the position vector field...

a. Show that the outward flux of the position vector field $\mathbf{F}=$ $x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$ through a smooth closed surface $S$ is three times the volume of the region enclosed by the surface. b. Let $\mathbf{n}$ be the outward unit normal vector field on $S$. Show that it is not possible for $\mathbf{F}$ to be orthogonal to $\mathbf{n}$ at every point of $S$.

a. Show that the outward flux of the position vector field $\mathbf{F}=$ $x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$ through a smooth closed surface $S$ is three times the volume of the region enclosed by the surface.
b. Let $\mathbf{n}$ be the outward unit normal vector field on $S$. Show that it is not possible for $\mathbf{F}$ to be orthogonal to $\mathbf{n}$ at every point of $S$.

Key Concepts

Unit Normal Vector and Orthogonality

The unit normal vector is a fundamental concept used to describe the direction perpendicular to a surface at each point. In flux calculations, it determines the orientation of the surface relative to the vector field. Demonstrating that the vector field cannot be orthogonal to the unit normal everywhere on a closed surface relies on the inherent contradiction that would arise—if true, it would imply a zero net outward flux, conflicting with the result obtained through the divergence theorem.

Flux Computation

Flux is the measure of the quantity of a vector field that passes through a surface. In this context, by applying the divergence theorem to a vector field with constant divergence, one can establish that the total outward flux is a scaled version of the volume of the region. This connection simplifies the evaluation of flux in problems involving closed surfaces.

Divergence Theorem

The divergence theorem is a key result in vector calculus that connects the flux of a vector field through a closed surface with the volume integral of its divergence over the region enclosed by the surface. It transforms complex surface integrals into simpler volume integrals, which is particularly useful when the divergence is constant, allowing for straightforward computation of the flux.

Vector Field Divergence

The divergence of a vector field quantifies the net rate of outflow of the field per unit volume at a given point. For a position vector field like (x, y, z), the divergence turns out to be a constant value, 3. This result, when combined with the divergence theorem, shows that the total outward flux through a closed surface is directly proportional to the volume of the enclosed region.

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