Given an irrotational vector field \overrightarrow{ F }=\left(k_{1} x y+k_{2} z^{3}\right) \vec{a}_{x}+\left(3 x^{2}-k_{3} z\right) \vec{a}_{y}+\left(3 x z^{2}-y\right) \vec{a}_{z} Find \nabla \overrightarrow{ F } at (1, 1, – 2).
Question 1.5.4: Given an irrotational vector field F→=(k1 xy+k2 z^3)a→x+(3x^...
Given
\begin{aligned}&\vec{F}=\left(k_{1} x y+k_{2} z^{3}\right) \hat{a}_{x}+\left(3 x^{2}-k_{3} z\right) \hat{a}_{y}+\left(3 y z^{2}-y\right) \hat{a}_{a} \\&\begin{aligned}\nabla \cdot F &=\frac{\partial}{\partial y} F_{x}+\frac{\partial}{\partial y} F_{y}+\frac{\partial}{\partial z} F_{z} \\&=k_{1} y+0+6 x z=k_{1} y+6 x z\end{aligned} \\&\text { at }(1,1,-2) \\&\nabla \cdot \vec{F}=k_{1}(1)+6(1)(-2)=k_{1}-12\end{aligned}Related Answered Questions
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