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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.

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