यदि  = 3xzî + 2xyĵ - yz² k̂ तो div \(\rm \vec v\) क्या है?

अवधारणा:

\(\rm div \rm \;\bar r =∇ .{\rm{\;\vec r\;}}\)

जहाँ ∇ = \(\rm \frac{{\partial }}{{\partial x}} \hat{i}+ \frac{{\partial }}{{\partial y }} \hat{j} +\frac{{\partial }}{{\partial z }} \hat{k}\)

गणना:

दिया हुआ: \(\rm \vec v \)= 3xzî + 2xyĵ - yz2 k̂

\(\text {div}\;\rm \rm \vec v=∇ .{\rm{\;\vec v\;}}\)

\(= (\rm \frac{{\partial }}{{\partial x}} \hat{i}+ \frac{{\partial }}{{\partial y }} \hat{j} +\frac{{\partial }}{{\partial z }} \hat{k}) \cdot \left( 3xz\hat{i}+ 2xy\hat{j} - yz^2\hat{k} \right) \)

\(= \rm \frac{{\partial (3xz) }}{{\partial x}} + \frac{{\partial (2xy)}}{{\partial y }} +\frac{{\partial (-yz^2)}}{{\partial z }}\\= \rm 3z\frac{{\partial (x) }}{{\partial x}} + 2y\frac{{\partial (x)}}{{\partial y }} -y\frac{{\partial (z^2)}}{{\partial z }}\)

= 3z + 2x - 2yz