GATE Civil 2023 Set 2 | Question: 1

Question

Let $\phi$ be a scalar field, and $\boldsymbol{u}$ be a vector field. Which of the following identities is true for $\operatorname{div}(\phi \boldsymbol{u})$?

 

• A. $\operatorname{div}(\phi \boldsymbol{u})=\phi \operatorname{div}(\boldsymbol{u})+\boldsymbol{u} \cdot \operatorname{grad}(\phi)$
• B. $\operatorname{div}(\phi \boldsymbol{u})=\phi \operatorname{div}(\boldsymbol{u})+\boldsymbol{u} \times \operatorname{grad}(\phi)$
• C. $\operatorname{div}(\phi \boldsymbol{u})=\phi \operatorname{grad}(\boldsymbol{u})+\boldsymbol{u} \cdot \operatorname{grad}(\phi)$
• D. $\operatorname{div}(\phi \boldsymbol{u})=\phi \operatorname{grad}(\boldsymbol{u})+\boldsymbol{u} \times \operatorname{grad}(\phi)$