Find the Conservative Vector Field for the Potential Function $h ( x , y , z ) = 16 x y \ln ( x + y )$

question 105

Multiple Choice

Find the conservative vector field for the potential function $h ( x , y , z ) = 16 x y \ln ( x + y )$ by finding its gradient.

• A) \(\mathbf { G } ( x , y , z ) = \left( \frac { 33 x y } { x + y } + 33 y \ln ( x + y ) \right) \mathbf { i } + \left( \frac { 33 x y } { x + y } + 33 x \ln ( x + y ) \right) \mathbf { j }\)
• B)\(\mathbf { G } ( x , y , z ) = \left( \frac { 770 x y } { x + y } + 770 y \ln ( x + y ) \right) \mathbf { i } + \left( \frac { 770 x y } { x + y } + 770 x \ln ( x + y ) \right) \mathbf { j }\)
• C)\(\mathbf { G } ( x , y , z ) = \left( \frac { 49 x y } { x + y } + 49 y \ln ( x + y ) \right) \mathbf { i } + \left( \frac { 49 x y } { x + y } + 49 x \ln ( x + y ) \right) \mathbf { j }\)
• D)\(\mathbf { G } ( x , y , z ) = \left( \frac { 16 x y } { x + y } + 16 y \ln ( x + y ) \right) \mathbf { i } + \left( \frac { 16 x y } { x + y } + 16 x \ln ( x + y ) \right) \mathbf { j }\)
• E)\(\mathbf { G } ( x , y , z ) = \left( \frac { 257 x y } { x + y } + 257 y \ln ( x + y ) \right) \mathbf { i } + \left( \frac { 257 x y } { x + y } + 257 x \ln ( x + y ) \right) \mathbf { j }\)

---

Definitions:

Return on Investment

An indicator for assessing an investment's effectiveness, determined by dividing the net profit by the investment's total cost.

Selling Price

The financial charge to a customer for purchasing a product or service.

Industrial Grinder

A heavy-duty machine used for reducing the size of materials through grinding or crushing in industrial settings.

Value-based Pricing

A pricing strategy where the price is based on the perceived value of a product or service to the customer rather than on the cost of production or historical prices.