Identifying invalid expressions: Let F and U be vector field,scalar field, resp'ly.

[WaleedEhsan04]

Hello,

Hope you are all having a good day. I am going over this question as revision and would like to find out if I am approaching it correctly. The question is the following:

Let F and U be a vector field and a scalar fieldrespectively. For each of the following expressions, state whether it is a scalar field(scalar), a vector field (vector) or an invalid expression (invalid):
• U^2(∇ × F) · (∇ · F)
• U(∇ · F)∇(∇ · F)
• ∇ × (∇U)
• ∇ × ∇(∇ · F)

Okay so for the first bullet point U^2 (∇ × F) · (∇ · F) my answer is: Vector Field
For the second bullet point U(∇ · F)∇(∇ · F) my answer is: Vector Field
For the third bullet point ∇ × (∇U) my answer is: Vector Field
For the fourth bullet point ∇ × ∇(∇ · F) my answer is: Vector Field

I know that it is unlikely that all of them are a vector field, but can anyone correct me if I am wrong? Any feedback would be appreciated.

Thank you

 

[Dr.Peterson]

Hello,

Hope you are all having a good day. I am going over this question as revision and would like to find out if I am approaching it correctly. The question is the following:

Let F and U be a vector field and a scalar field respectively. For each of the following expressions, state whether it is a scalar field(scalar), a vector field (vector) or an invalid expression (invalid):
• U^2(∇ × F) · (∇ · F)
• U(∇ · F)∇(∇ · F)
• ∇ × (∇U)
• ∇ × ∇(∇ · F)

Okay so for the first bullet point U^2 (∇ × F) · (∇ · F) my answer is: Vector Field
For the second bullet point U(∇ · F)∇(∇ · F) my answer is: Vector Field
For the third bullet point ∇ × (∇U) my answer is: Vector Field
For the fourth bullet point ∇ × ∇(∇ · F) my answer is: Vector Field

I know that it is unlikely that all of them are a vector field, but can anyone correct me if I am wrong? Any feedback would be appreciated.

Thank you

Can you explain your thinking for each answer, rather than just the answer? Just state, step by step as you would evaluate an expression, what kind of object you have -- e.g. is ∇ × F a scalar or a vector? How about ∇ · F? Then, can you take the dot product of those? What do you get? And so on.

 

[HallsofIvy]

Hello,

Hope you are all having a good day. I am going over this question as revision and would like to find out if I am approaching it correctly. The question is the following:

Let F and U be a vector field and a scalar fieldrespectively. For each of the following expressions, state whether it is a scalar field(scalar), a vector field (vector) or an invalid expression (invalid):
• U^2(∇ × F) · (∇ · F)
• U(∇ · F)∇(∇ · F)
• ∇ × (∇U)
• ∇ × ∇(∇ · F)

Okay so for the first bullet point U^2 (∇ × F)·(∇ · F)my answer is: Vector Field

What is the "·" between "(∇ × F)" and "(∇ · F)" supposed to mean? You have used it elsewhere as the dot product and wrote scalar multiplication as juxtaposition. But "(∇ · F)" is not a vector.

For the second bullet point U(∇ · F)∇(∇ · F) my answer is: Vector Field

For the this one, both "(∇ · F)" and "∇(∇ · F)" are vectors but you have neither "·", "×" between them.

For the third bullet point ∇ × (∇U) my answer is: Vector Field

Yes, for U a scalar field ∇U is a vector field and its curl is a vector field.

For the fourth bullet point ∇ × ∇(∇ · F) my answer is: Vector Field

Yes, ∇ · F is a scalar field so ∇(∇ · F) is a vector field and ∇ × ∇(∇ · F) is a vector field.

I know that it is unlikely that all of them are a vector field, but can anyone correct me if I am wrong? Any feedback would be appreciated.

Thank you