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Chapter 14: Problem 7

Why does a two-dimensional vector field with zero divergence on a region have zero outward flux across a closed curve that bounds the region?

Short Answer

Expert verified
Question: Prove that a two-dimensional vector field with zero divergence on a region has zero outward flux across a closed curve that bounds the region. Answer: Utilizing the two-dimensional divergence theorem, which states that the outward flux through a closed curve is equal to the integral of the divergence within the enclosed region, and given that the divergence is zero everywhere in the region, we find that the outward flux across the closed curve is also zero. This proves that a two-dimensional vector field with zero divergence on a region results in zero outward flux across a closed curve that bounds the region.

Step by step solution

01

Define the given variables

Let's denote the given two-dimensional vector field as F(x, y) and the closed curve bounding the region as C.
02

Recall the divergence theorem for a two-dimensional case

In two-dimensional case, the divergence theorem relates the flux of a vector field through a closed curve to the divergence of the field inside the region enclosed by the curve. The divergence theorem states that: \(鈭甠{C} F \cdot n ds = 鈭琠{R}divF dA\) Here, F = two-dimensional vector field F(x, y), n = outward unit normal vector to the curve C, ds = infinitesimal arc length element along the curve C, R = the region enclosed by the curve C, divF = divergence of F, and dA = infinitesimal area element in the region R.
03

Interpret the given condition

It is given that the divergence of the vector field F is zero everywhere in the region. So, we have: \(divF = 0\)
04

Apply the divergence theorem

Now, we use the two-dimensional divergence theorem to find the relationship between the flux and the divergence: \(鈭甠{C} F \cdot n ds = 鈭琠{R}divF dA\) Since \(divF = 0\), we get: \(鈭甠{C} F \cdot n ds = 鈭琠{R}(0)dA\) Now we can integrate on both sides: \(鈭甠{C} F \cdot n ds = 0\) This shows that the outward flux across the closed curve C is zero. Thus, it is proved that a two-dimensional vector field with zero divergence on a region has zero outward flux across a closed curve that bounds the region.

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