Problem 59 - Thomas Calculus

Question

Circulation Find the circulation of \(\mathbf{F}=2 x \mathbf{i}+2 z \mathbf{j}+2 y \mathbf{k}\) around the closed path consisting of the following three curves traversed in the direction of increasing \(t\) . \begin{equation} \begin{array}{ll}{C_{1} :} & {\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq \pi / 2} \\ {C_{2} :} & {\mathbf{r}(t)=\mathbf{j}+(\pi / 2)(1-t) \mathbf{k}, \quad 0 \leq t \leq 1} \\\ {C_{3} :} & {\mathbf{r}(t)=t \mathbf{i}+(1-t) \mathbf{j}, \quad 0 \leq t \leq 1}\end{array} \end{equation}

Step-by-Step Solution

Answer

Total circulation is 0.

Step 1: Understand the Definition of Circulation

Circulation is the line integral of a vector field over a closed curve. Mathematically, it’s given by the integral \( \oint_C \mathbf{F} \cdot d\mathbf{r} \), where \( \mathbf{F} \) is the vector field and \( d\mathbf{r} \) is the differential element along the curve \( C \). This requires breaking down the closed path into segments if composed of multiple curves.

Step 2: Calculate the Line Integral for Curve \( C_1 \)

For the curve \( C_1: \mathbf{r}(t) = (\cos t) \mathbf{i} + (\sin t) \mathbf{j} + t \mathbf{k} \) where \( 0 \leq t \leq \pi/2 \), compute the derivative \( \mathbf{r}'(t) = (-\sin t) \mathbf{i} + (\cos t) \mathbf{j} + \mathbf{k} \). Substitute \( \mathbf{r}(t) \) into \( \mathbf{F} \), giving \( \mathbf{F}(t) = 2\cos t \mathbf{i} + 2t \mathbf{j} + 2\sin t \mathbf{k} \). The line integral over \( C_1 \) is \( \int_0^{\pi/2} \mathbf{F}(t) \cdot \mathbf{r}'(t) \, dt \). Calculate this integral: \( = \int_0^{\pi/2} [-2\cos t \sin t + 2t \cos t + 2\sin t] \, dt \). Evaluate this integral.

Step 3: Calculate the Line Integral for Curve \( C_2 \)

For the curve \( C_2: \mathbf{r}(t) = \mathbf{j} + (\pi/2)(1-t) \mathbf{k} \) where \( 0 \leq t \leq 1 \), compute the derivative \( \mathbf{r}'(t) = 0 \mathbf{i} + 0 \mathbf{j} - (\pi/2) \mathbf{k} \). Substitute \( \mathbf{r}(t) \) into \( \mathbf{F} \), finding \( \mathbf{F}(t) = 0 \mathbf{i} + 2 \mathbf{j} \). The line integral over \( C_2 \) is \( \int_0^1 \mathbf{F}(t) \cdot \mathbf{r}'(t) \, dt = \int_0^1 [2 \cdot 0 - 0 \cdot (\pi/2)] \, dt \). Evaluate this integral (result is 0).

Step 4: Calculate the Line Integral for Curve \( C_3 \)

For the curve \( C_3: \mathbf{r}(t) = t \mathbf{i} + (1-t) \mathbf{j} \) where \( 0 \leq t \leq 1 \), compute the derivative \( \mathbf{r}'(t) = \mathbf{i} - \mathbf{j} + 0\mathbf{k} \). Substitute \( \mathbf{r}(t) \) into \( \mathbf{F} \), finding \( \mathbf{F}(t) = 2t \mathbf{i} \). The line integral over \( C_3 \) is \( \int_0^1 \mathbf{F}(t) \cdot \mathbf{r}'(t) \, dt = \int_0^1 [2t \cdot 1 + 0 - 0] \, dt \). Evaluate this integral.

Step 5: Sum the Integrals for Total Circulation

Add the results from Steps 2, 3, and 4 to find the total circulation over the entire closed path \( C_1 + C_2 + C_3 \). This is \( \oint_C \mathbf{F} \cdot d\mathbf{r} = (\text{Result of Step 2}) + 0 + 1 = (\text{final sum}) \).

Key Concepts

Line Integral

A line integral is a way to integrate functions along a curve, incorporating both the function's value and the distance along the path. It’s crucial in fields like physics and engineering to calculate work done by a force or the circulation of a vector field. In our context, we calculate the line integral of the vector field \( \mathbf{F} \) along a specified curve \( C \). This requires us to break down the path into smaller segments, especially if the path consists of multiple curves.

To compute a line integral in vector fields, we follow these steps:

• Parameterize the curve \( C \) by a vector function \( \mathbf{r}(t) \).
• Compute the derivative of \( \mathbf{r}(t) \), \( \mathbf{r}'(t) \), which gives the direction of the curve at each point.
• Substitute \( \mathbf{F}(\mathbf{r}(t)) \) into the line integral expression \( \int \mathbf{F} \cdot d\mathbf{r} \).
• Calculate the dot product \( \mathbf{F}(t) \cdot \mathbf{r}'(t) \) and integrate over the given limits of \( t \).

This line integral evaluates how the vector field acts along the path, whether it adds to the motion or works against it. Understanding and computing the line integral effectively helps in determining the vector field's behavior in specified regions.

Vector Calculus

Vector calculus is a branch of mathematics that helps describe physical phenomena like fluid flow and electromagnetism using calculus and vectors. It is especially useful in analyzing vector fields and performing various operations on them, including differentiation and integration along curves and surfaces.

In vector calculus, the most common operations include:

• Gradient - Measures the rate and direction of change in a scalar field.
• Divergence - Provides the rate of change of density of a vector field.
• Curl - Describes the rotation or swirling strength of a field.

These tools allow us to solve complex real-world problems by converting them into mathematical expressions that can be analyzed. Understanding vector calculus is key to controlling and predicting dynamic systems.

Closed Path Integration

Closed path integration involves integrating a function around a closed loop or path. This concept is fundamental when determining the circulation of a vector field around a loop, which describes how a field "flows" or circulates around the path.

A critical aspect of closed path integration is:

• It's often denoted by the integral symbol with a circle \( \oint \), indicating the path is closed.
• Stokes' theorem and the fundamental theorem for line integrals are essential here. They connect the line integral around a closed path to the curl of the vector field over the surface bounded by the path.

These mathematical frameworks help compute values that might appear complex by breaking them down into more manageable parts. Understanding closed path integration is crucial for fields dealing with continuity and flow, like fluid dynamics and electrical circuits.