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

Question

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

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Consider the definition of the curl of a vector field in two dimensions. ‖ ‖ 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.⚹◊What is the formula for the curl of a vector field in two dimensions?⍭ Generate me a similar question◊

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