The vector function \[\overrightarrow {\text{A}} \] is given by \[\overrightarrow {\text{A}} = \overrightarrow \nabla {\text{u,}}\] where u(x, y) is a scalar function, Then \[\left| {\overrightarrow \nabla \times \overrightarrow {\text{A}} } \right|\] A. -1 B. 0 C. 1 D. \[\infty \]

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

The gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and its magnitude is equal to the magnitude of the greatest rate of increase of the scalar field.

The curl of a vector field is a vector field that represents the circulation of the vector field around a small loop.

The curl of the gradient of a scalar field is always zero. This is because the gradient points in the direction of the greatest rate of increase of the scalar field, and the curl measures the circulation of a vector field around a small loop. If the vector field is always pointing in the same direction, then there is no circulation around a small loop, and the curl is zero.

In this case, the vector field $\overrightarrow {\text{A}}$ is given by $\overrightarrow {\text{A}} = \overrightarrow \nabla {\text{u,}}$ where u(x, y) is a scalar function. This means that $\overrightarrow {\text{A}}$ is the gradient of the scalar field u(x, y). Therefore, the curl of $\overrightarrow {\text{A}}$ is zero.