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In my fluidum mechanics course we encounter a lot of vector calculus problems, one of which I am struggling with for a while now. We must prove that: $$ \vec{V} \cdot (\vec{\nabla}\vec{V}) = (\vec{\nabla}\cdot\vec{V})\vec{V} $$ solely using sommation/index notation. $\vec{\nabla}\vec{V}$ is a second order tensor which we denote by: $$\left(\sum\limits_{i}\hat{e_i}\frac{\partial}{\partial x_i}\right)\left(\sum\limits_{j}\hat{e_j}v_j\right) $$ I think my confusion lies in the use of $\frac{\partial}{\partial x_i}$ in a tensor since we haven't used tensors commonly before taking this course. Could someone maybe prove this and clarify how second order tensors work in general?

In my fluid mechanics course we encounter a lot of vector calculus problems, one of which I have been struggling with for a while now. We must prove that $$ \vec{V} \cdot \left(\vec{\nabla}\vec{V}\right) = \left(\vec{\nabla}\cdot\vec{V}\right)\vec{V} $$ solely using summation/index notation. $\vec{\nabla}\vec{V}$ is a second order tensor which we denote by: $$\left(\sum\limits_{i}\hat{e}_{i}\frac{\partial}{\partial x_i}\right)\left(\sum\limits_{j}\hat{e}_{j}V_j\right).$$ I think my confusion lies in the use of $\frac{\partial}{\partial x_i}$ in a tensor since we haven't used tensors commonly before taking this course. Could someone maybe prove this and clarify how second order tensors work in general?

In my fluidum mechanics course we encounter a lot of vector calculus problems, one of which I am struggling with for a while now. We must prove that: $$ \vec{V} \cdot (\vec{\nabla}\vec{V}) = (\vec{\nabla}\cdot\vec{V})\vec{V} $$ solely using sommation/index notation. $\vec{\nabla}\vec{V}$ is a second order tensor which we denote by: $$(\sum\limits_{i}\hat{e_i}\frac{\partial}{\partial x_i})(\sum\limits_{j}\hat{e_j}v_j) $$ I think my confusion lies in the use of $\frac{\partial}{\partial x_i}$ in a tensor since we haven't used tensors commonly before taking this course. Could someone maybe prove this and clarify how second order tensors work in general?

In my fluidum mechanics course we encounter a lot of vector calculus problems, one of which I am struggling with for a while now. We must prove that: $$ \vec{V} \cdot (\vec{\nabla}\vec{V}) = (\vec{\nabla}\cdot\vec{V})\vec{V} $$ solely using sommation/index notation. $\vec{\nabla}\vec{V}$ is a second order tensor which we denote by: $$\left(\sum\limits_{i}\hat{e_i}\frac{\partial}{\partial x_i}\right)\left(\sum\limits_{j}\hat{e_j}v_j\right) $$ I think my confusion lies in the use of $\frac{\partial}{\partial x_i}$ in a tensor since we haven't used tensors commonly before taking this course. Could someone maybe prove this and clarify how second order tensors work in general?

In my fluidum mechanics course we encounter a lot of vector calculus problems, one of which I am struggling with for a while now. We must prove that: $$ \vec{V} \cdot (\vec{\nabla}\vec{V}) = (\vec{\nabla}\cdot\vec{V})\vec{V} $$ soley using sommation/index notation. $\vec{\nabla}\vec{V}$ is a second order tensor which we denote by: $$(\sum\limits_{i}\hat{e_i}\frac{\partial}{\partial x_i})(\sum\limits_{j}\hat{e_j}v_j) $$ I think my confusion lies in the use of $\frac{\partial}{\partial x_i}$ in a tensor since we haven't used tensors commonly before taking this course. Could someone maybe proof this and clarify how second order tensors work in general?

In my fluidum mechanics course we encounter a lot of vector calculus problems, one of which I am struggling with for a while now. We must prove that: $$ \vec{V} \cdot (\vec{\nabla}\vec{V}) = (\vec{\nabla}\cdot\vec{V})\vec{V} $$ solely using sommation/index notation. $\vec{\nabla}\vec{V}$ is a second order tensor which we denote by: $$(\sum\limits_{i}\hat{e_i}\frac{\partial}{\partial x_i})(\sum\limits_{j}\hat{e_j}v_j) $$ I think my confusion lies in the use of $\frac{\partial}{\partial x_i}$ in a tensor since we haven't used tensors commonly before taking this course. Could someone maybe prove this and clarify how second order tensors work in general?