Question : Find divergence of the following Vector Field
\[F\left(x,y,z\right)=\left(xyz\ ,\ -yz^2,\ x^2\right)\]
Solution :
To Find divergence of Vector Field \[F\left(x,y,z\right)=\left(xyz\ ,\ -yz^2,\ x^2z\right)\] Writing this as vector \[F\left(x,y,z\right)=xyz\hat{i}-yz^2\hat{j}+x^2z\hat{k}\] Now divergence of Vector Field F(x,y,z) is given as \[\nabla.\ F=\left(\hat{i}\frac{\partial}{\partial x}+\hat{j}\frac{\partial}{\partial y}+\hat{k}\frac{\partial}{\partial z}\right).\left(xyz\hat{i}-yz^2\hat{j}+x^2z\hat{k}\ \right)\] \[\nabla.\ F=\hat{i}\frac{\partial}{\partial x}\left(xyz\right)+\hat{j}\frac{\partial}{\partial y}\left(-yz^2\right)+\hat{k}\frac{\partial}{\partial z}\left(x^2z\right)\] \[\nabla.\ F=\hat{i}\left(yz\right)+\hat{j}\left({-z}^2\right)+\hat{k}\left(x^2\right)\] \[\nabla.\ F=yz\hat{i}-z^2\hat{j}+x^2\hat{k}\] Is the required divergence of given Vector Field F(x,y,z).