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Magazine Chapter 5: Problem 28
Compute the circulation and net flux for the given flow and the indicated closed contour \(C\). \(f(z)=\bar{z} ;\) where \(C\) is the square with vertices \(z=0, z=1, z=1+i, z=i\)
Short Answer
Expert verified
The circulation is \(2i + 1\) and the net flux is 0.
Step by step solution
01
Identify the Flow Function and Contour
The flow function is given by \(f(z) = \bar{z}\), which is the complex conjugate of \(z\). The contour \(C\) is a square with vertices at \(0, 1, 1+i, i\).
02
Define the Parametric Equations
The square contour can be divided into four path segments: 1. From \(0\) to \(1\), 2. From \(1\) to \(1+i\), 3. From \(1+i\) to \(i\), 4. From \(i\) to \(0\). Parametrize each path for analysis.
03
Calculate Circulation Around Each Segment
For each segment, express \(\bar{z}\) in terms of the parameterization variable and integrate over the path. For example, for the first segment from \(0\) to \(1\), the path can be parametrized as \(z(t) = t\) where \(t\) ranges from 0 to 1.
04
Compute the Circulation Along Segment 1
Parametrize the first segment as \(z(t) = t\), where \(0 \leq t \leq 1\). The conjugate \(\bar{z} = t\).The circulation is \[\int_0^1 \bar{z}(t) \, dt = \int_0^1 t \, dt = \left[ \frac{t^2}{2} \right]_0^1 = \frac{1}{2}.\]
05
Compute the Circulation Along Segment 2
Parametrize the second segment as \(z(t) = 1 + it\), where \(0 \leq t \leq 1\). The conjugate \(\bar{z} = 1 - it\).The circulation is \[\int_0^1 (1-it) \, i dt = \int_0^1 (i - t) \, dt = \left[ it - \frac{t^2}{2} \right]_0^1 = i - \frac{1}{2}.\]
06
Compute the Circulation Along Segment 3
Parametrize the third segment as \(z(t) = (1-t) + i\), where \(0 \leq t \leq 1\). The conjugate \(\bar{z} = (1-t) - i\).The circulation is \[\int_0^1 ((1-t) - i) \, -i dt = \int_0^1 ((1-t) - i) \, (-i) dt = \left[ t + it \right]_0^1 = 1 + i.\]
07
Compute the Circulation Along Segment 4
Parametrize the fourth segment as \(z(t) = it\), where \(0 \leq t \leq 1\). The conjugate \(\bar{z} = -it\).The circulation is \[\int_0^1 -it \, -1 dt = \int_0^1 t \, dt = \left[ \frac{t^2}{2} \right]_0^1 = \frac{1}{2}.\]
08
Add Contributions to Find Total Circulation
Add the circulations from all four segments: \[\frac{1}{2} + (i - \frac{1}{2}) + (1 + i) + \frac{1}{2} = 2i + 1.\]Thus, the total circulation around the contour \(C\) is \(2i + 1\).
09
Examine the Cauchy-Riemann Equations
The given function \(f(z) = \bar{z}\) does not satisfy the Cauchy-Riemann equations, indicating that the divergence must be recalculated because it contains non-analytic properties. This affects the net flux.
10
Calculate the Divergence for Net Flux
For \(f(z) = x - iy\), the divergence is given by \(abla \cdot f = 0\), since both partial derivatives \(\frac{\partial}{\partial x}(x)\) and \(\frac{\partial}{\partial y}(-y)\) add to zero.
11
Determine the Net Flux
Since the divergence of the function is zero, the net flux across any closed contour in the plane is zero.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Circulation
Circulation in complex analysis refers to the line integral of a vector field along a closed contour. It helps us understand how a fluid flows around a particular path, and in this context, circulation measures the total amount of flow that encircles a contour. In our exercise, we are given a flow function \(f(z) = \bar{z}\), which represents the complex conjugate of \(z\). The contour is a square with specific vertices.
- The contour's path is made up of four segments, each going from one vertex to another.
- To find the circulation, we integrate the complex conjugate function along these segments.
- Specifically, we are looking at the path integral of \(\bar{z}\) along these segments.
- Summing up the results from these integrations gives us the total circulation for the square contour.
Net Flux
Net flux measures how much of a vector field flows through a surface or an area. More specifically, while circulation is about flow around a path, net flux is concerned with flow through a surface. In mathematical terms, it's often linked to the divergence of the vector field.
For the given function \(f(z) = \bar{z}\), the exercise states that the divergence is zero.
For the given function \(f(z) = \bar{z}\), the exercise states that the divergence is zero.
- The divergence of a vector field \( abla \cdot f\) indicates how much the field spreads out from a point.
- In this context, the divergence is calculated using the components of the function.
- Because the components \( x - iy\) satisfy a divergence of zero, the flow does not spread out.
Cauchy-Riemann Equations
The Cauchy-Riemann equations are fundamental in complex analysis, dictating when a complex function is differentiable (or analytic). These equations help determine if a function can be expressed as a derivative in the complex plane.
The equations state that for a function \(f(z) = u(x, y) + iv(x, y)\) to be analytic:
The equations state that for a function \(f(z) = u(x, y) + iv(x, y)\) to be analytic:
- The partial derivatives must satisfy \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y\) as well as \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\).
- This implies the function is non-analytic and doesn't satisfy the Cauchy-Riemann equations.
- Because of this, we should be cautious with properties like differentiability and the presence of poles.
Complex Conjugate
A complex conjugate is a fundamental concept when dealing with complex numbers. For any complex number \(z = x + yi\), its complex conjugate is \(\bar{z} = x - yi\). It reflects the original complex number about the real axis on the complex plane.
This relationship has several properties that make it useful:
This relationship has several properties that make it useful:
- When a complex number is multiplied by its conjugate, it results in a real number, \(z \cdot \bar{z} = x^2 + y^2\).
- It provides a way to negate the imaginary part, which can be helpful in solving complex equations.
- Since the path integral involves the conjugate function \(\bar{z}\), it impacts both the real and imaginary components of the flow calculations.
- Understanding the properties of conjugation is essential when solving problems in complex analysis, as it affects integral results and theoretical interpretations.
