Final answer:
The curl of the gradient of a scalar field results in a zero vector, hence the first statement is false. The gradient of the divergence of a vector field is a meaningful operation, making the second statement also false.
Step-by-step explanation:
If F is a vector field and f is a scalar field, then the curl of the gradient of f is indeed meaningless and results in a zero vector. This is a property of vector calculus known as the curl of a gradient being always zero, which can be mathematically expressed as ∇ × (∇f) = 0. Hence, the answer to the first part of the question is false.
For the second part of the question, the gradient of the divergence of F is meaningful. It is a second-order derivative operation on vector fields, which produces a vector field. This operation is often used in physics, particularly in the context of fluid dynamics and electromagnetism. Therefore, the answer to the second part is false.