দেওয়া আছে, $$ \begin{aligned} \vec{A}&=2x^2\hat{i}-3yz\hat{j}+x^2\hat{k} \\ \varphi&=2z-x^2y \end{aligned} $$ প্রথমে $\vec{\nabla}\varphi$ নির্ণয় করি: $$ \begin{aligned} \vec{\nabla}\varphi&=\left(\frac{\partial}{\partial x}\hat{i}+\frac{\partial}{\partial y}\hat{j}+\frac{\partial}{\partial z}\hat{k}\right)(2z-x^2y) \\ &=\frac{\partial}{\partial x}(2z-x^2y)\hat{i} + \frac{\partial}{\partial y}(2z-x^2y)\hat{j} + \frac{\partial}{\partial z}(2z-x^2y)\hat{k} \\ &=(-2xy)\hat{i} + (-x^2)\hat{j} + (2)\hat{k} \end{aligned} $$ এখন, $(1, -1, 1)$ বিন্দুতে $\vec{A}$ এবং $\vec{\nabla}\varphi$ এর মান বের করি: $$ \begin{aligned} \vec{A}_{(1,-1,1)}&=2(1)^2\hat{i}-3(-1)(1)\hat{j}+(1)^2\hat{k} = 2\hat{i}+3\hat{j}+\hat{k} \\ \vec{\nabla}\varphi_{(1,-1,1)}&=-2(1)(-1)\hat{i}-(1)^2\hat{j}+2\hat{k} = 2\hat{i}-\hat{j}+2\hat{k} \end{aligned} $$ অতএব, ডট গুণফল: $$ \begin{aligned} \vec{A} \cdot \vec{\nabla}\varphi&=(2\hat{i}+3\hat{j}+\hat{k}) \cdot (2\hat{i}-\hat{j}+2\hat{k}) \\ &=(2)(2) + (3)(-1) + (1)(2) \\ &=4-3+2 \\ &=3 \end{aligned} $$ নির্ণেয় মান $3$।