যদি \( \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \) হয়, তবে \( \nabla \cdot \vec{r} \) কত?

যদি \( \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \) হয়, তবে \( \nabla \cdot \vec{r} \) নির্ণয় করতে হবে। আমরা জানি, \( \nabla = \frac{\partial}{\partial x}\hat{i} + \frac{\partial}{\partial y}\hat{j} + \frac{\partial}{\partial z}\hat{k} \) সুতরাং, \( \nabla \cdot \vec{r} = \left( \frac{\partial}{\partial x}\hat{i} + \frac{\partial}{\partial y}\hat{j} + \frac{\partial}{\partial z}\hat{k} \right) \cdot \left( x\hat{i} + y\hat{j} + z\hat{k} \right) \) ডট গুণনের নিয়ম অনুসারে, \( \nabla \cdot \vec{r} = \frac{\partial}{\partial x}(x) + \frac{\partial}{\partial y}(y) + \frac{\partial}{\partial z}(z) \) যেহেতু, \( \frac{\partial}{\partial x}(x) = 1 \), \( \frac{\partial}{\partial y}(y) = 1 \) এবং \( \frac{\partial}{\partial z}(z) = 1 \), তাই \( \nabla \cdot \vec{r} = 1 + 1 + 1 = 3 \) অতএব, \( \nabla \cdot \vec{r} = 3 \) 🎉