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

WaleedEhsan04

New member
Joined
May 25, 2018
Messages
1
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

Elite Member
Joined
Nov 12, 2017
Messages
5,542
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

Elite Member
Joined
Jan 27, 2012
Messages
5,433
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
 
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