**Example Problem: Calculating Work Done by a Force Field**In this example, we will determine the work done by a given force field $ \mathbf{F} $ as it moves an object from one point to another in a three-dimensional space.### The Problem StatementThe force field $ \mathbf{F} $ moves an object from the initial point $ (0, 0, 1) $ to the final point $ (2, 1, 0) $. The force field is given by the vector function:\[\mathbf{F}(x,y,z) = (x - y^2)\mathbf{i} + (y - z^2)\mathbf{j} + (z - x^2)\mathbf{k}\]We need to find the total work done by the force field $ \mathbf{F} $ over this path.### Multiple Choice AnswersThe potential answers for the total work done are:a. $\frac{7}{2} $b. $\frac{7}{9} $c. $\frac{7}{5} $d. $\frac{7}{4} $e. $\frac{7}{3} $### Detailed Solution Steps- **Vector Function $ \mathbf{F} $:** \[ \mathbf{F}(x,y,z) = (x - y^2)\mathbf{i} + (y - z^2)\mathbf{j} + (z - x^2)\mathbf{k} \]- **Initial and Final Points:** Initial point: $ (0, 0, 1) $ Final point: $ (2, 1, 0) $- **Evaluate the Work Done $ W $:** The work done by the force field can be derived through integration of the dot product of the force vector $ \mathbf{F} $ and the displacement vector $ \mathbf{dr} $ along the given path. Since the specific path may vary, there is often a simplifying assumption that the path involves straight-line movement between the two points where possible, unless specified otherwise.In general, the formula for calculating the work done is:\[W = \int_{C} \mathbf{F} \cdot d\mathbf{r}\]where \( \mathbf{

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Answered: The force field F moves an object from…

The force field F moves an object from (0, 0, 1) to (2, 1, 0) F(x,v,z) = (x - y?)i + (y – z?)j + (z – x?)k Find the total work done by F ? а. 7/2 b. 7/9 с. 7/5 d. 7/4 е. 7/3

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

**Example Problem: Calculating Work Done by a Force Field**

In this example, we will determine the work done by a given force field $ \mathbf{F} $ as it moves an object from one point to another in a three-dimensional space.

### The Problem Statement

The force field $ \mathbf{F} $ moves an object from the initial point $ (0, 0, 1) $ to the final point $ (2, 1, 0) $. The force field is given by the vector function:

\[
\mathbf{F}(x,y,z) = (x - y^2)\mathbf{i} + (y - z^2)\mathbf{j} + (z - x^2)\mathbf{k}
\]

We need to find the total work done by the force field $ \mathbf{F} $ over this path.

### Multiple Choice Answers

The potential answers for the total work done are:
a. $\frac{7}{2} $
b. $\frac{7}{9} $
c. $\frac{7}{5} $
d. $\frac{7}{4} $
e. $\frac{7}{3} $

### Detailed Solution Steps

- **Vector Function $ \mathbf{F} $:**

\[
\mathbf{F}(x,y,z) = (x - y^2)\mathbf{i} + (y - z^2)\mathbf{j} + (z - x^2)\mathbf{k}
\]

- **Initial and Final Points:**

Initial point: $ (0, 0, 1) $

Final point: $ (2, 1, 0) $

- **Evaluate the Work Done $ W $:**

The work done by the force field can be derived through integration of the dot product of the force vector $ \mathbf{F} $ and the displacement vector $ \mathbf{dr} $ along the given path.

Since the specific path may vary, there is often a simplifying assumption that the path involves straight-line movement between the two points where possible, unless specified otherwise.

In general, the formula for calculating the work done is:

\[
W = \int_{C} \mathbf{F} \cdot d\mathbf{r}
\]

where \( \mathbf{

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