যদি  vecr=xhati+yhatj+zhatk  হয়, তবে  vecnabla.vecr  কত ?

Explanation:

Another Explanation (5): যদি \(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\) হয়, তবে \(\vec{\nabla} \cdot \vec{r}\) এর মান নির্ণয় করতে হবে। আমরা জানি, \(\vec{\nabla} = \frac{\partial}{\partial x}\hat{i} + \frac{\partial}{\partial y}\hat{j} + \frac{\partial}{\partial z}\hat{k}\) তাহলে, \(\vec{\nabla} \cdot \vec{r} = \left(\frac{\partial}{\partial x}\hat{i} + \frac{\partial}{\partial y}\hat{j} + \frac{\partial}{\partial z}\hat{k}\right) \cdot \left(x\hat{i} + y\hat{j} + z\hat{k}\right)\) ডট গুণনের নিয়ম অনুযায়ী, \(\vec{\nabla} \cdot \vec{r} = \frac{\partial}{\partial x}(x) + \frac{\partial}{\partial y}(y) + \frac{\partial}{\partial z}(z)\) যেহেতু \(\frac{\partial}{\partial x}(x) = 1\), \(\frac{\partial}{\partial y}(y) = 1\) এবং \(\frac{\partial}{\partial z}(z) = 1\), সুতরাং, \(\vec{\nabla} \cdot \vec{r} = 1 + 1 + 1 = 3\) 🎉 অতএব, \(\vec{\nabla} \cdot \vec{r} = 3\) 🥳