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If a vector field is conservative, it’s true.

Let $\textbf{F}:\mathbb{R}^3 \to \mathbb{R}^3$ be a vector field that we want to show is conservative, that is, its curl is zero.

Firstly, what would be the representation of a vector field?

My attempt would be:
$$\textbf{F}(x, y, z) = F_1(x, y, z)\textbf{i} + F_2(x, y, z)\textbf{j} + F_3(x, y, z)\textbf{k}$$
And since the curl is given by:
$$\operatorname{curl}\textbf{F} = \operatorname{curl}(F_1 \textbf{i} + F_2 \textbf{j} + F_3 \textbf{k}) = \operatorname{curl}(F_1 \textbf{i} + F_2 \textbf{j} + F_3 \textbf{k}) = \operatorname{curl}(F_1 \textbf{i}) + \operatorname{curl}(F_2 \textbf{j}) + \operatorname{curl}(F_3 \textbf{k}) = 0$$
Is it correct?

A:

Yes, you’re good. That’s how you’d represent a vector field.
You’re not wrong. Your curl calculation is correct. However, \$\oper

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