Question

A vector field \({\bf{F}}\)is shown. Use the interpretation of divergence derived in this section to determine whether div \({\bf{F}}\)is positive or negative at \({P_1}\)and at \({P_2}\).

Short answer

It is derived that in this section \({\bf{F}}(x,y) = 0\) for those \((x,y)\) that lie on the line \(y = 2x\).

Step-by-step solution

Statement of divergence theorem

The divergence theorem is a mathematical expression of the physical truth that, in the absence of matter creation or destruction, the density within a region of space can only be changed by allowing it to flow into or out of the region through its boundary.

Plot the vector field

Consider the given information.

\({\bf{F}} = \left( {{y^2} - 2xy} \right){\bf{i}} + \left( {3xy - 6{x^2}} \right){\bf{j}}\)

Rewrite this as: \({\bf{F}} = (y - 2x)(y{\bf{i}} + 3x{\bf{j}})\)

Note that the vector field is zero at the points which lie on the line \(y - 2x = 0\,.\)Magnitude of the vectors increase as we move away from the line \(y - 2x = 0\,.\)

Now plot the vector field.

 Find \((x,y)\) such that \({\bf{F}}(x,y) = 0\)

Let us solve the problem.

Now, find \((x,y)\) such that \({\bf{F}}(x,y) = 0\).

\(\nabla f = \left( {x{e^{xy}} \cdot y + {e^{xy}},{x^2}{e^{xy}}} \right)\)

Therefore, \({\bf{F}}(x,y) = 0\) for those \((x,y)\) that lie on the line \(y = 2x\).