Find the Gradient Field of the Function $$f ( x , y , z ) = \frac { x z + x y + y z } { x y z }$$

question 228

Multiple Choice

Find the gradient field of the function.
- $$f ( x , y , z ) = \frac { x z + x y + y z } { x y z }$$

• A) \({\nabla} f = \frac { 1 } { \mathrm { x } ^ { 2 } } \mathrm { i } + \frac { 1 } { \mathrm { y } ^ { 2 } } \mathrm { j } + \frac { 1 } { \mathrm { z } ^ { 2 } } \mathbf { k }\)
• B) \({\nabla} f = - \frac { 1 } { \mathrm { x } ^ { 2 } } \mathrm { i } - \frac { 1 } { \mathrm { y } ^ { 2 } } \mathrm { j } - \frac { 1 } { \mathrm { z } ^ { 2 } } \mathrm { k }\)
• C) \({\nabla} f = - \frac { 1 } { x ^ { 2 } y z } \mathbf { i } - \frac { 1 } { x y _ { y } ^ { 2 } z } \mathbf { j } - \frac { 1 } { x y z ^ { 2 } } \mathbf { k }\)
• D) \({\nabla} f =\frac{1}{x^{2} y z} i+\frac{1}{x y^{2} z} j+\frac{1}{x y z^{2}} k\)

---

Definitions:

Density

In a general context, it refers to the measure of mass per unit volume; in marketing, it may relate to the saturation of advertisements or products in a market area.

Geographical Area

A specific part of the earth's surface defined by natural or artificial boundaries.

Manufacturer-Sponsored Wholesale Franchise System

A form of franchising where the manufacturer licenses distributors to sell its products to retailers, maintaining control over branding and marketing.

Soft Drink Industry

Refers to the sector involving the production, marketing, and sale of non-alcoholic beverages, such as soda, packaged water, and juices.