Find the Gradient Vector Field of the Scalar Function $f$ (That Is, Find the Conservative Vector Field

question 36

Multiple Choice

Find the gradient vector field of the scalar function $f$ . (That is, find the conservative vector field $\mathbf { F }$ for the potential function $f$ of $\mathbf { F }$ .)
$$f ( x , y , z ) = 7 x y ^ { 2 } + 2 y z ^ { 2 }$$

• A) \(\mathbf { F } ( x , y , z ) = \left( 7 y ^ { 2 } \right) \mathbf { i } + \left( 2 z ^ { 2 } \right) \mathbf { j } + ( 4 y z ) \mathbf { k }\)
• B) \(\mathbf { F } ( x , y , z ) = ( 14 x y ) \mathbf { i } + ( 14 x y ) \mathbf { j } + \left( 2 z ^ { 2 } \right) \mathbf { k }\)
• C) \(\mathbf { F } ( x , y , z ) = \left( 7 y ^ { 2 } \right) \mathbf { i } + \left( 14 x y + 2 z ^ { 2 } \right) \mathbf { j } + ( 4 y z ) \mathbf { k }\)
• D) \(\mathbf { F } ( x , y , z ) = \left( 7 y ^ { 2 } \right) \mathbf { i } + ( 14 x y ) \mathbf { j } + ( 4 y z ) \mathbf { k }\)

---

Definitions:

Income Statement

A report detailing a business's income, expenditures, and profit for a certain timeframe.

Gross Profit

Revenue from sales minus the cost of goods sold before administrative, financial, and other operating expenses are deducted.

Stockholders' Equity

The residual interest in the assets of a corporation after deducting its liabilities, indicating the ownership value.

Income From Operations

The profit realized from a business's operational activities, calculated before taxes and interest charges.