Let $$\nabla \cdot \left( {{\text{f}}\overrightarrow {\text{v}} } \right) = {{\text{x}}^2}{\text{y}} + {{\text{y}}^2}{\text{z}} + {{\text{z}}^2}{\text{x}},$$ where f and v are scalar and vector fields respectively. If $$\overrightarrow {\text{v}} = {\text{y}}\overrightarrow {\text{i}} + {\text{z}}\overrightarrow {\text{j}} + {\text{x}}\overrightarrow {\text{k}} ,$$ then $$\overrightarrow {\text{v}} \cdot \nabla {\text{f}}$$ is A. x2y + y2z + z2x B. 2xy + 2yz + 2zx C. x + y + z D. 0

The correct answer is $\boxed{0}$.

The divergence of a product of a scalar field and a vector field is the sum of the product of the scalar field and the divergence of the vector field, plus the vector field times the curl of the scalar field. In this case, the scalar field is $f$ and the vector field is $\mathbf{v}$. The divergence of $\mathbf{v}$ is $0$, since $\mathbf{v}$ is a constant vector field. The curl of $f$ is also $0$, since $f$ is a scalar field. Therefore, the divergence of the product of $f$ and $\mathbf{v}$ is $0$.

Here is a more detailed explanation of each option:

• Option A: $x^2y + y^2z + z^2x$. This is the product of the scalar field $f$ and the vector field $\mathbf{v}$. However, the divergence of a product of a scalar field and a vector field is the sum of the product of the scalar field and the divergence of the vector field, plus the vector field times the curl of the scalar field. In this case, the scalar field is $f$ and the vector field is $\mathbf{v}$. The divergence of $\mathbf{v}$ is $0$, since $\mathbf{v}$ is a constant vector field. The curl of $f$ is also $0$, since $f$ is a scalar field. Therefore, the divergence of the product of $f$ and $\mathbf{v}$ is $0$, which is not equal to $x^2y + y^2z + z^2x$.
• Option B: $2xy + 2yz + 2zx$. This is the curl of the vector field $\mathbf{v}$. However, the divergence of a product of a scalar field and a vector field is the sum of the product of the scalar field and the divergence of the vector field, plus the vector field times the curl of the scalar field. In this case, the scalar field is $f$ and the vector field is $\mathbf{v}$. The divergence of $\mathbf{v}$ is $0$, since $\mathbf{v}$ is a constant vector field. The curl of $f$ is also $0$, since $f$ is a scalar field. Therefore, the divergence of the product of $f$ and $\mathbf{v}$ is $0$, which is not equal to $2xy + 2yz + 2zx$.
• Option C: $x + y + z$. This is the sum of the components of the vector field $\mathbf{v}$. However, the divergence of a product of a scalar field and a vector field is the sum of the product of the scalar field and the divergence of the vector field, plus the vector field times the curl of the scalar field. In this case, the scalar field is $f$ and the vector field is $\mathbf{v}$. The divergence of $\mathbf{v}$ is $0$, since $\mathbf{v}$ is a constant vector field. The curl of $f$ is also $0$, since $f$ is a scalar field. Therefore, the divergence of the product of $f$ and $\mathbf{v}$ is $0$, which is not equal to $x + y + z$.
• Option D: $0$. This is the correct answer. The divergence of a product of a scalar field and a vector field is the sum of the product of the scalar field and the divergence of the vector field, plus the vector field times the curl of the scalar field. In this case, the scalar field is $f$ and the vector field is $\mathbf{v}$. The divergence of $\mathbf{v}$ is $0$, since $\mathbf{v}$ is a constant vector field. The curl of $f$ is also $0$, since $f$ is a scalar field. Therefore, the divergence of the product of $f$ and $\mathbf{v}$ is $0$.