Question

Math

Posted 3 months ago

```
Let $F$ be a 2D vector field.
Is the expression $\operatorname{grad}(\operatorname{curl}(F))$ a scalar field, a vector field, or undefined?
Choose 1 answer:
(A) Scalar field
(B) Vector field
(C) Undefined
```

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Answer from Sia

Posted 3 months ago

Solution by Steps

step 2

The curl of a 2D vector field $F$ is a scalar function, not a vector

step 3

The gradient (denoted as $\operatorname{grad}$) of a scalar function is a vector field

step 4

Therefore, the expression $\operatorname{grad}(\operatorname{curl}(F))$ takes the curl of $F$, which is a scalar, and then computes the gradient of that scalar, resulting in a vector field

B

Key Concept

Curl and Gradient in Vector Calculus

Explanation

In vector calculus, the curl of a vector field in two dimensions results in a scalar function. Taking the gradient of this scalar function then results in a vector field. Thus, the expression $\operatorname{grad}(\operatorname{curl}(F))$ is defined and results in a vector field.

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