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:

Effective Communication

The process of exchanging information in a way that ensures the message is received and understood as intended.

Efficient

Achieving maximum productivity with minimum wasted effort or expense.

Efficient Communication

The exchange of information in a way that is effective and consumes the least amount of resources possible.

Minimum Cost

The least amount of financial expenditure required to achieve a particular objective, often essential for budgeting and financial planning.