A force is described by the vector field (mathbf{F}=-alphaleft(9 x^{2} hat{mathbf{i}}+2 z^{2} hat{mathbf{j}}+6 y z hat{mathbf{k}} ight)). After obtaining the dimensions of (alpha), verify that

A force is described by the vector field \(\mathbf{F}=-\alpha\left(9 x^{2} \hat{\mathbf{i}}+2 z^{2} \hat{\mathbf{j}}+6 y z \hat{\mathbf{k}}\right)\). After obtaining the dimensions of \(\alpha\), verify that the field is irrotational. Determine the potential energy function \(V\) and calculate the work \(W\) done by the force to bring an object from the point \((1,2,-1)\) to the point \((0,1,1)\).