Problem 51 - 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 \(t\) $$ \begin{array}{ll}{C_{1} :} & {\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq \pi / 2} \\ {C_{2} :} & {\mathbf{r}(t)=\mathbf{j}+(\pi / 2)(1-t) \mathbf{k}, \quad 0 \leq t \leq 1} \\\ {C_{3} :} & {\mathbf{r}(t)=t \mathbf{i}+(1-t) \mathbf{j}, \quad 0 \leq t \leq 1}\end{array} $$

Step-by-Step Solution

Answer

The total circulation is \( \pi/2 + 1 \).

Step 1: Parameterize the Curves

Identify the parametric equations for the curves. We have:- For \( C_1: \) \( \mathbf{r}_1(t) = (\cos t) \mathbf{i} + (\sin t) \mathbf{j} + t \mathbf{k} \) with \( 0 \leq t \leq \pi/2 \).- For \( C_2: \) \( \mathbf{r}_2(t) = \mathbf{j} + (\pi/2)(1-t) \mathbf{k} \) with \( 0 \leq t \leq 1 \).- For \( C_3: \) \( \mathbf{r}_3(t) = t \mathbf{i} + (1-t) \mathbf{j} \) with \( 0 \leq t \leq 1 \).

Step 2: Calculate \( d\mathbf{r} \) for Each Curve

Find the differential \( d\mathbf{r} \) for each curve:- For \( C_1: \) \( d\mathbf{r}_1 = (-\sin t) dt \mathbf{i} + (\cos t) dt \mathbf{j} + dt \mathbf{k} \).- For \( C_2: \) \( d\mathbf{r}_2 = 0 \mathbf{i} + 0 \mathbf{j} - (\pi/2) dt \mathbf{k} \).- For \( C_3: \) \( d\mathbf{r}_3 = dt \mathbf{i} - dt \mathbf{j} + 0 \mathbf{k} \).

Step 3: Evaluate the Line Integral over Each Curve

Compute the line integral \( \int_{C} \mathbf{F} \cdot d\mathbf{r} \) for each segment:- For \( C_1: \) the expression \( \mathbf{F} \cdot d\mathbf{r}_1 = 2\cos t (-\sin t) + 2t (\cos t) + 2\sin t dt = 2t\cos t dt \).Perform the integration from \( 0 \) to \( \pi/2 \):\[ \int_0^{\pi/2} 2t\cos t \, dt \]Using integration by parts, this evaluates to \( \pi/2 \).- For \( C_2: \) since \( d\mathbf{r}_2 \) is entirely in the \( \mathbf{k} \) direction and \( \mathbf{F} \cdot \mathbf{k} = 0 \), the integral is 0.- For \( C_3: \) \( \mathbf{F} \cdot d\mathbf{r}_3 = 2t dt \).Perform the integration from \( 0 \) to \( 1 \):\[ \int_0^{1} 2t \, dt = 1 \].

Step 4: Sum of Integrals for Total Circulation

Add the integral values from each segment to find the total circulation:- From \( C_1: \) \( \pi/2 \)- From \( C_2: \) 0- From \( C_3: \) 1The total circulation is \( \pi/2 + 0 + 1 = \pi/2 + 1 \).

Key Concepts

Vector Fields

A vector field can be thought of as a map that assigns a vector to each point in space. Imagine standing on a terrain where, at each point, the wind has a specific direction and strength. This idea can be visually similar to how vector fields are depicted in mathematics and physics. In our problem, the vector field is given by \( \mathbf{F}=2x \mathbf{i} + 2z \mathbf{j} + 2y \mathbf{k} \). Here, \( x, y, \) and \( z\) are the coordinates in three-dimensional space, and \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are the unit vectors in the direction of the x, y, and z axes, respectively.

Vector fields are critical in applications like fluid dynamics or electromagnetism, where the flow of a substance or field is analyzed over a region. The main concern is often how a vector field interacts with a path or line within the field, like finding the circulation or the total 'flowing around' a closed loop.

Parametric Equations

Parametric equations allow us to describe a path or curve in space using a single parameter, often denoted as \( t \). It simplifies complex curves by breaking them into components that can be expressed separately. For instance, in our exercise, we have three curves defined using parametric forms:

• For \( C_1 \): \( \mathbf{r}_1(t) = (\cos t) \mathbf{i} + (\sin t) \mathbf{j} + t \mathbf{k} \)
• For \( C_2 \): \( \mathbf{r}_2(t) = \mathbf{j} + (\pi/2)(1-t) \mathbf{k} \)
• For \( C_3 \): \( \mathbf{r}_3(t) = t \mathbf{i} + (1-t) \mathbf{j} \)

Here, these parametric equations help us follow each segment of the closed path individually, which is crucial when calculating line integrals.

The parameter \( t \) usually varies over a specific interval. By changing the value of \( t \), you trace out the entire curve, offering a powerful method to visualize and work with different paths efficiently.

Calculus Problems

Calculus allows us to deal with changing quantities and their accumulation. In tasks like computing line integrals, we bring calculus together with vector fields and parametric equations to solve practical problems.

Line integrals, specifically, help compute the total effect of a vector field along a path. It involves taking a smooth path and summing up infinitesimal contributions to find the total effect. In our exercise, we compute line integrals like \( \int_{C_1} \mathbf{F} \cdot d\mathbf{r}_1 \) over distinct curve segments \( C_1, C_2, \text{ and } C_3 \). These integrals essentially capture how much of the vector field 'flows through' each segment of the path.

To calculate a line integral, we first need to differentiate the parametric equations to find \( d\mathbf{r} \) for each curve, which represents infinitesimal path vectors. Then, we compute \( \mathbf{F} \cdot d\mathbf{r} \), integrate over the given parameter interval, and add results from each path.

This integration results in a total value, such as the circulation, which portrays how a field interacts with a path. Calculus problems often demand precise attention to detail but ultimately uncover fascinating behaviors in physics and engineering.