Step 1

a.

The circulation of the vector field F on C is the net component of F in the direction tangent to C and it can be defined as \(\displaystyle\int_{{c}}{F}\cdot{d}{r}\).

The object experiences different vector force at a different point of the vector field.

The double integral of a quantity, iver the region R, measures the rotation at each point of R.

Therefore, the circulation of object around curve C is the result of the rotation of vector field inside region R.

So, the given statement is true.

Step 2

b.

The two dimensional divergence is \(\displaystyle\frac{{{d}{f}}}{{{\left.{d}{x}\right.}}}+\frac{{{d}{g}}}{{{\left.{d}{y}\right.}}}\).

The two-dimensional divergence results in flux across the region, not for the circulation.

So, if divergence is 0, then the flux is 0, not circulation.

So, the given statement is false.

Step 3

c.

The two dimensional curl is \(\displaystyle\frac{{{d}{g}}}{{{\left.{d}{x}\right.}}}-\frac{{{d}{f}}}{{{\left.{d}{y}\right.}}}\).

Evaluation of curl results in the circulation over the curve.

So, if the curve is positive, then the circulation is also positive.

So, the given statement is true.

a.

The circulation of the vector field F on C is the net component of F in the direction tangent to C and it can be defined as \(\displaystyle\int_{{c}}{F}\cdot{d}{r}\).

The object experiences different vector force at a different point of the vector field.

The double integral of a quantity, iver the region R, measures the rotation at each point of R.

Therefore, the circulation of object around curve C is the result of the rotation of vector field inside region R.

So, the given statement is true.

Step 2

b.

The two dimensional divergence is \(\displaystyle\frac{{{d}{f}}}{{{\left.{d}{x}\right.}}}+\frac{{{d}{g}}}{{{\left.{d}{y}\right.}}}\).

The two-dimensional divergence results in flux across the region, not for the circulation.

So, if divergence is 0, then the flux is 0, not circulation.

So, the given statement is false.

Step 3

c.

The two dimensional curl is \(\displaystyle\frac{{{d}{g}}}{{{\left.{d}{x}\right.}}}-\frac{{{d}{f}}}{{{\left.{d}{y}\right.}}}\).

Evaluation of curl results in the circulation over the curve.

So, if the curve is positive, then the circulation is also positive.

So, the given statement is true.