Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The work required to move an

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The work required to move an object around a closed curve C in the presence of a vector force field is the circulation of the force field on the curve.
b. If a vector field has zero divergence throughout a region (on which the conditions of Green’s Theorem are met), then the circulation on the boundary of that region is zero.
c. If the two-dimensional curl of a vector field is positive throughout a region (on which the conditions of Green’s Theorem are met), then the circulation on the boundary of that region is positive (assuming counterclockwise orientation).

Answers (1)

2020-10-28
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.
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