Problem 64 - Thomas Calculus

Question

In Exercises \(63-68,\) use a CAS to perform the following steps for finding the work done by force F over the given path: $$\begin{array}{l}{\mathbf{F}=\frac{3}{1+x^{2}} \mathbf{i}+\frac{2}{1+y^{2}} \mathbf{j} ; \quad \mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}} \\\ {0 \leq t \leq \pi}\end{array}$$

Step-by-Step Solution

Answer

The work done by the force \( \mathbf{F} \) along the given path is the evaluated integral value.

Step 1: Understand the Scenario

We are given a force vector \( \mathbf{F} = \frac{3}{1+x^{2}} \mathbf{i} + \frac{2}{1+y^{2}} \mathbf{j} \) and a path represented by \( \mathbf{r}(t) = (\cos t) \mathbf{i} + (\sin t) \mathbf{j} \), with the parameter \( t \) ranging from \( 0 \) to \( \pi \). The task is to find the work done by this force along the given path, calculated via the line integral of \( \mathbf{F} \cdot d \mathbf{r} \).

Step 2: Calculate the Differential Element of the Path

The differential element of the path, \( d \mathbf{r} \), can be found by differentiating \( \mathbf{r}(t) \) with respect to \( t \). This results in \( d \mathbf{r} = -\sin t\, dt \, \mathbf{i} + \cos t \, dt \, \mathbf{j} \).

Step 3: Substitute the Parametric Equations into \( \mathbf{F} \)

Substitute \( x = \cos t \) and \( y = \sin t \) into \( \mathbf{F} \) to get \( \mathbf{F} = \frac{3}{1+\cos^2 t} \mathbf{i} + \frac{2}{1+\sin^2 t} \mathbf{j} \).

Step 4: Set Up the Line Integral

The work done is given by the integral \( \int_{0}^{\pi} \mathbf{F} \cdot d \mathbf{r} \). This means we need to compute \( \int_{0}^{\pi} \left( \frac{3}{1 + \cos^2 t}(- \sin t) + \frac{2}{1 + \sin^2 t}(\cos t) \right) \, dt \).

Step 5: Solve the Integral

Evaluate the integral \( \int_{0}^{\pi} \left( \frac{3}{1 + \cos^2 t}(- \sin t) + \frac{2}{1 + \sin^2 t}(\cos t) \right) \, dt \) using a Computer Algebra System (CAS) to find the exact value of the work done.

Step 6: Interpret the Result

The CAS provides the value of the integral, which is the total work done by the force \( \mathbf{F} \) along the specified path. This number represents the amount of energy transferred by the force.

Key Concepts

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector fields and differentiable forms. It's crucial in physics and engineering to understand quantities that have both magnitude and direction, such as force, velocity, and acceleration.
One main topic within vector calculus is the line integral. The line integral of a vector field along a path calculates accumulated quantities, like work or flux, along that path.
This concept is illustrated well in the original exercise, where we are tasked to find the work done by a force vector along a curved path. In this context, vector calculus helps to articulate complex relationships and interactions between vectors and specific paths.
Through this powerful tool, we can abstract and compute real-world applications such as the work done by a moving force or the flow of a fluid around an object.

Force Vector

A force vector is a mathematical entity in physics used to represent a force which has both magnitude and direction.
In the original exercise, the force vector is given by \( \mathbf{F} = \frac{3}{1+x^{2}} \mathbf{i} + \frac{2}{1+y^{2}} \mathbf{j} \). This expresses the force components in the Cartesian coordinate system. Here, \( \mathbf{i} \) and \( \mathbf{j} \) are unit vectors along the x and y-axis respectively.
A proper understanding of the force vector is vital.

• The components of the vector provide information on how the force is acting along each axis.
• The vector gets substituted with parametric variables later, making it adaptable over different paths.

Manipulating this vector is central to solving problems related to work and dynamics, leading directly into calculations involving vector calculus and line integrals.

Parametric Equations

Parametric equations are equations where a set of quantities are expressed as explicit functions of a common variable, known as the parameter.
In the given exercise, the path of the motion is specified by a parametric equation \( \mathbf{r}(t) = (\cos t) \mathbf{i} + (\sin t) \mathbf{j} \), which provides a way to trace the path using the parameter \( t \).
Parametric equations help in:

• Describing curved paths, which aren't possible with simple Cartesian equations.
• Allowing complex movement to be broken down into simpler linear components along chosen parameter intervals.

By turning paths into detailed steps through parametric equations, we can plug these directly into other formulas and solve integrals more accurately, particularly when dealing with forces and movements in vector calculus.

Computer Algebra System

A Computer Algebra System (CAS) is a sophisticated software tool that performs symbolic mathematics.
CAS like Mathematica, Maple, or MATLAB are often used to solve complex integrals and derivatives that would otherwise require tedious manual steps and are prone to human error.
Using these systems in vector calculus enables:

• Precise calculations of integrals and derivatives.
• Handling otherwise intractable algebraic expressions efficiently.

In the exercise, a CAS is applied to evaluate the line integral, which provides the exact value for the work done. The system saves time and ensures accuracy, offering a quick interpretation and solution to problems involving complex expressions. Leveraging CAS technologies is an advantageous approach in advanced mathematics, science, and engineering.