Determine Whether $\mathbf { F }$ Is Conservative $f$ Such That $\mathbf { F } = \nabla f$

question 35

Short Answer

Determine whether $\mathbf { F }$ is conservative. If so, find a function $f$ such that $\mathbf { F } = \nabla f$ .
$$\mathbf { F } ( x , y , z ) = 4 x ^ { 3 } y ^ { 2 } z ^ { 3 } \mathbf { i } + 2 x ^ { 4 } y z ^ { 3 } \mathbf { j } + 3 x ^ { 4 } y ^ { 2 } z ^ { 2 } \mathbf { k }$$

---

Definitions:

Debtor

An individual or entity that owes a debt to another; the opposite party being a creditor.

Secured Creditor

A creditor that has a legal claim, called a lien, on the debtor's assets, ensuring priority payment if the debtor defaults on the obligation.

Unsecured Creditor

A creditor who extends credit without requiring specific collateral to secure the debt owed by the borrower.

Collateral

Assets pledged by a borrower to secure a loan or credit, which can be seized if the loan is not repaid.