Determine Whether or Not $\mathbf { F } ( x , y ) = \left( y e ^ { x y } + 4 x ^ { 3 } y \right) \mathbf { i } + \left( x e ^ { x y } + x ^ { 4 } \right) \mathbf { j }$

question 152

Essay

Determine whether or not $\mathbf { F } ( x , y ) = \left( y e ^ { x y } + 4 x ^ { 3 } y \right) \mathbf { i } + \left( x e ^ { x y } + x ^ { 4 } \right) \mathbf { j }$ is a conservative vector field. If it is, find a function f such that $\mathbf { F } = \nabla f$ .

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Definitions:

Cost Of Goods Sold

In a perpetual inventory system, an account that records the cost of merchandise inventory used to make the sale. Also, it is the total cost of the goods which were sold to customers.

Finished Goods Inventory

Products that are completed and ready for sale but still remain in the company's inventory.

Cost Of Goods Manufactured

The total cost incurred to produce goods within a specific period, including materials, labor, and overhead.

Cost Of Goods Sold

The direct costs attributable to the production of the goods sold by a company, including both materials and direct labor costs.