Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 7: Problem 11

Let \(f(z)=u+i v\) be differentiable. Consider the vector field given by \(\mathbf{F}=v \mathbf{i}+u \mathbf{j}\). Show that the equations \(\nabla \cdot \mathbf{F}=\mathbf{0}\) and \(\nabla \times \mathbf{F}=\mathbf{0}\) are equivalent to the Cauchy-Riemann Equations. [You will need to recall from multivariable calculus the del operator, \(\left.\nabla=\mathbf{i} \frac{\partial}{\partial x}+\mathrm{j} \frac{\partial}{\partial y}+\mathbf{k} \frac{\partial}{\partial z} \cdot\right]\)

Short Answer

Expert verified
The calculation of the divergence and curl of vector field \(\mathbf{F}=v \mathbf{i}+u \mathbf{j}\) under the conditions \(\nabla \cdot \mathbf{F} = 0\) and \(\nabla \times \mathbf{F} = 0\) gives system of equations \( \frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y} \) and \(\frac{\partial u}{\partial x} =\frac{\partial v}{\partial y} \) respectively. These equations correspond to the Cochy-Riemann equations, proving that they are equivalent.

Step by step solution

01

Compute the Divergence of \(\mathbf{F}\)

First, compute the divergence (\(\nabla \cdot \mathbf{F}\)) of the vector field \(\mathbf{F}\). This involves taking the dot product of the del operator with \(\mathbf{F}\). Note that the z-components of the del operator and \(\mathbf{F}\) are both zero, so they won't contribute to the result. Using \(\mathbf{F}=v \mathbf{i}+u \mathbf{j}\), the divergence is given by \( \nabla \cdot \mathbf{F} = \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \). Setting this to zero (as per the exercise) gives \(\frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y}\).
02

Compute the Curl of \(\mathbf{F}\)

Next, compute the curl (\(\nabla \times \mathbf{F}\)) of the vector field \(\mathbf{F}\). The curl involves calculating the determinant of a 3x3 matrix in which the first row contains the unit vectors, the second row contains the del operator and the third row contains the vector field. Given that the z-components of the del operator and \(\mathbf{F}\) are both zero, the determinant simplifies to: \( (\frac{\partial u}{\partial x} - \frac{\partial v}{\partial y}) \mathbf{k} \). Setting this to zero gives \(\frac{\partial u}{\partial x} =\frac{\partial v}{\partial y} \).
03

Relate to the Cauchy-Riemann Equations

Correlate the results obtained from steps 1 and 2 with the Cauchy-Riemann equations:\( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \(\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \). It can be seen that both pairs of equalities match and hence, the respective equations \(\nabla \cdot \mathbf{F} = 0\) and \(\nabla \times \mathbf{F} = 0\) are indeed equivalent to the Cauchy-Riemann equations.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the Laurent series expansion for \(f(z)=\frac{\sinh z}{2^{2}}\) about \(z=0\). [Hint You need to first do a MacLaurin series expansion for the hyperbolic sine.]

Write the following in polar form, \(z=r e^{i \theta}\). a. \(i-1\). b. \(-2 i\). c. \(\sqrt{3}+3 i\).

Show that \(\sin (x+i y)=\sin x \cosh y+i \cos x \sinh y\) using trigonometric identities and the exponential forms of these functions.

Find the principal value of \(i^{i}\). Rewrite the base, \(i\), as an exponential first.

Show that $$ \int_{C} \frac{d z}{(z-1-i)^{n+1}}=\left\\{\begin{array}{cl} 0, & n \neq 0 \\ 2 \pi i, & n=0 \end{array}\right. $$ for \(C\) the boundary of the square \(0 \leq x \leq 2,0 \leq y \leq 2\) taken counterclockwise. [Hint: Use the fact that contours can be deformed into simpler shapes (like a circle) as long as the integrand is analytic in the region between them. After picking a simpler contour, integrate using parametrization.].
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free