\( \nabla \cdot \vec{r} \) নির্ণয়

দেওয়া আছে, \( \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \)

ডাইভার্জেন্স অপারেটর, \( \nabla = \frac{\partial}{\partial x}\hat{i} + \frac{\partial}{\partial y}\hat{j} + \frac{\partial}{\partial z}\hat{k} \)

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 অনুযায়ী,

\( \nabla \cdot \vec{r} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} \)

আমরা জানি, \( \frac{\partial x}{\partial x} = 1 \), \( \frac{\partial y}{\partial y} = 1 \) এবং \( \frac{\partial z}{\partial z} = 1 \)

অতএব, \( \nabla \cdot \vec{r} = 1 + 1 + 1 = 3 \) 🎉

সুতরাং, \( \nabla \cdot \vec{r} = 3 \) 😊