Question

To determine,

a) To explain \(\int_C F \cdot dr\) is positive, negative or zero.

b) To explain\({\mathop{\rm div}\nolimits} F(P)\) is positive, negative or zero.

Short answer

1. The value of \(\int_C F \cdot dr\) is negative and explained.
2. The value of \({\mathop{\rm div}\nolimits} F(P)\) is positive and explained.

Step-by-step solution

Given data

The figure of vector field \({\bf{F}}\), a curve \(C\), and a point \(P\) is given.

Concept of Stokes theorem

The classical Stokes' theorem can be stated in one sentence.

Thelineintegralof a vector field over a loop is equal to thefluxof its curlthrough the enclosed surface.

Explanation for the value of \(\int_C F   \cdot dr\)

a)

In given figure, the vector starting on point \(C\) directed in opposite direction to \(C\), so the tangential component \(F \cdot T\) is negative.

Represent \(dr\) in terms of \(Tds\).

\(dr = Tds\)

Find the value of \(\int_C F \cdot dr\).

\(\int_C F \cdot dr = \int_C F \cdot Tds\)

Since, \(F \cdot T\) is negative, the value of \(\int_C F \cdot Tds\) is negative.

\(\int_C F \cdot dr = - \int_C F \cdot Tds\)

Thus, the value of \(\int_C F \cdot dr\) is negative and explained.

Explanation of the value of \({\mathop{\rm div}\nolimits} F(P)\)

b)

In given figure, the vectors that starts near the point \(P\) is longer than vectors that ends near the point \(P\), so the net flow is in outward direction near point \(P\) and hence \({\mathop{\rm div}\nolimits} F(P)\) is positive.

Thus, the value of \({\mathop{\rm div}\nolimits} F(P)\) is positive and explained.