Problem 49 - Thomas Calculus

Question

\(\mathbf{F}\) is the velocity field of a fluid flowing through a region in space. Find the flow along the given curve in the direction of increasing \(t .\) $$ \begin{array}{l}{\mathbf{F}=(x-z) \mathbf{i}+x \mathbf{k}} \\\ {\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{k}, \quad 0 \leq t \leq \pi}\end{array} $$

Step-by-Step Solution

Answer

The flow along the curve is \( \pi \).

Step 1: Understand the Velocity Field and Curve

Given the velocity field \( \mathbf{F} = (x-z) \mathbf{i} + x \mathbf{k} \) and the curve \( \mathbf{r}(t) = (\cos t) \mathbf{i} + (\sin t) \mathbf{k} \), we need to evaluate the flow along the curve in the direction of increasing \( t \) from \( 0 \) to \( \pi \).

Step 2: Parameterize the Curve

The curve is parameterized by \( \mathbf{r}(t) = (\cos t, 0, \sin t) \). It represents a path in the \( xz \)-plane with \( y = 0 \) for \( 0 \leq t \leq \pi \).

Step 3: Calculate the Derivative of the Curve

Determine the tangent vector for the curve by differentiating \( \mathbf{r}(t) \) with respect to \( t \):\[\mathbf{r}'(t) = \frac{d}{dt} [(\cos t) \mathbf{i} + (\sin t) \mathbf{k}] = (-\sin t) \mathbf{i} + (\cos t) \mathbf{k}.\]

Step 4: Substitute the Parameters into the Velocity Field

Substitute \( x = \cos t \), \( y = 0 \), and \( z = \sin t \) into \( \mathbf{F} \):\[\mathbf{F}(t) = ((\cos t) - (\sin t)) \mathbf{i} + (\cos t) \mathbf{k}.\]

Step 5: Compute the Dot Product Along the Curve

Calculate the dot product \( \mathbf{F}(t) \cdot \mathbf{r}'(t) \):\[((\cos t) - (\sin t))(-\sin t) + (\cos t)(\cos t) = -(\cos t \sin t) + \sin^2 t + \cos^2 t.\]

Step 6: Simplify the Expression

Use the identity \( \sin^2 t + \cos^2 t = 1 \) to simplify the expression:\[1 - \cos t \sin t.\]

Step 7: Integrate to Find the Flow

Integrate the simplified expression over the interval \( t = 0 \) to \( t = \pi \):\[\int_0^{\pi} (1 - \cos t \sin t) \, dt.\] Apply integration techniques:For \( \int_0^{\pi} 1 \, dt = \pi \).For \( \int_0^{\pi} \cos t \sin t \, dt \), use \( \cos t \sin t = \frac{1}{2} \sin(2t) \):\[\int_0^{\pi} \frac{1}{2} \sin(2t) \, dt = -\frac{1}{4} [\cos(2t)]_0^{\pi} = -\frac{1}{4}((-1) - 1) = 0.\]Thus, the total integrals sum to \( \pi \).

Step 8: Conclude the Result

The overall flow along the curve, taking into account both components of the field and path, is \( \pi \).

Key Concepts

Velocity Field

A velocity field is a vector field that represents the velocity of a fluid at every point in space. It's a fundamental concept in fluid dynamics, allowing us to visualize and analyze how a fluid moves over time. Here, the velocity field is given by the function \[ \mathbf{F} = (x-z) \mathbf{i} + x \mathbf{k} \]This equation tells us how the velocity changes in the spatial region. The components of this vector field represent velocities in the direction of the axes. The part \((x-z) \mathbf{i}\) indicates the velocity in the x-direction while \(x \mathbf{k}\) represents the velocity in the z-direction. Understanding this helps to know how fluid flows interact with objects or boundaries.

Parameterized Curve

In vector calculus, a parameterized curve is a way to represent a geometric curve using a parameter, often denoted by \(t\). It helps in describing a path in space, and it's particularly useful when understanding how fields, like velocity fields, interact with paths. In this exercise, the parameterized curve is \[ \mathbf{r}(t) = (\cos t) \mathbf{i} + (\sin t) \mathbf{k} \]covering the interval \(0 \leq t \leq \pi\).

• This equation describes a circular path in the xz-plane.
• Here, the parameter \(t\) usually represents time, indicating how the curve is traced out as time progresses.
• By substituting \(t\) values into \(\mathbf{r}(t)\), one can determine specific points on the curve.

This parameterization is essential when evaluating how a field is affected by the curve as it moves through it.

Dot Product

The dot product, also known as the scalar product, is an algebraic operation that occurs between two vectors. It yields a scalar quantity and has numerous applications in physics and engineering, such as determining the angle between vectors or calculating work done by a force. Mathematically, if you have two vectors, \(\mathbf{A}\) and \(\mathbf{B}\), their dot product is given by:\[ \mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z \]In this exercise, the dot product computes how the velocity field \(\mathbf{F}(t)\) interacts with the tangent vector \(\mathbf{r}'(t)\) of the curve. The computed dot product \[ ((\cos t) - (\sin t))(-\sin t) + (\cos t)(\cos t) = 1 - \cos t \sin t \]is crucial in evaluating the effect of the field along the path, providing an expression needed for further integration.

Integration

Integration is a mathematical process of finding the integral of a function, which can provide valuable insights in physics, such as calculating total quantities like displacement, area, or, in this case, fluid flow along a path. It allows accumulation over an interval, which is key when analyzing continuous changes like fluid motion.

• The integral computed here is \[\int_0^{\pi} (1 - \cos t \sin t) \, dt\]
• This breaks down into manageable parts using integration techniques.
• Evaluating \(\int_0^{\pi} 1 \, dt\) gives \(\pi\), and resolving \(\int_0^{\pi} \cos t \sin t \, dt\) by employing trigonometric identities results in 0.
• The final result provides the total flow along the path, determined by evaluating these integrals together.

Integration is vital in determining the cumulative effect of the velocity field around the path, summarizing the overall flow.