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Chapter 2: Problem 15

Calculate the curl and the divergence of each of the following vector fields. If the curl turns out to be zero, try to discover a scalar function \(\phi\) of which the vector field is the gradient: (a) \(F_{x}=x+y ; F_{y}=-x+y ; F_{z}=-2 z\). (b) \(G_{x}=2 y ; G_{y}=2 x+3 z ; G_{z}=3 y\). (c) \(H_{x}=x^{2}-z^{2} ; H_{y}=2 ; H_{z}=2 x z\).

Short Answer

Expert verified
Curl of (a) is zero, \(\phi = \frac{x^2}{2} - xy + \frac{y^2}{2} - z^2\). Curl of (b) is zero. Curl of (c) is \((0, -2z, 0)\).

Step by step solution

01

Understand the objective

The task is to calculate the curl and divergence of vector fields. The curl applies only to three-dimensional vectors and measures rotation, while divergence measures the rate of change of density. If the curl is zero, check if the vector is a gradient of a scalar function.
02

Calculate curl and divergence for (a)

For vector field \( \mathbf{F} = (x+y, -x+y, -2z) \), calculate the curl using \( abla \times \mathbf{F} \):\[\left( \frac{\partial}{\partial y}(-2z) - \frac{\partial}{\partial z}(-x+y), \frac{\partial}{\partial z}(x+y) - \frac{\partial}{\partial x}(-2z), \frac{\partial}{\partial x}(-x+y) - \frac{\partial}{\partial y}(x+y) \right)\]Evaluating each, get \( abla \times \mathbf{F} = (0, 0, 0) \). For divergence:\[abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(x+y) + \frac{\partial}{\partial y}(-x+y) + \frac{\partial}{\partial z}(-2z) = 1 + 1 - 2 = 0\].
03

Determine if \( \mathbf{F} \) is a gradient field

Since curl is zero, attempt to find a scalar function \( \phi(x, y, z) \) such that \( abla \phi = (x+y, -x+y, -2z) \). Integrate each gradient component: \( \frac{\partial \phi}{\partial x} = x+y \rightarrow \phi = \frac{x^2}{2} + xy + f(y,z) \). Similarly find \( \phi = -xy + \frac{y^2}{2} + g(x,z) \) and \( \phi = -z^2 + h(x,y) \). Reconciling the results gives \( \phi = \frac{x^2}{2} - xy + \frac{y^2}{2} - z^2 \).
04

Calculate curl and divergence for (b)

For vector field \( \mathbf{G} = (2y, 2x+3z, 3y) \), calculate:\[abla \times \mathbf{G} = \left( \frac{\partial}{\partial y}(3y) - \frac{\partial}{\partial z}(2x+3z), \frac{\partial}{\partial z}(2y) - \frac{\partial}{\partial x}(3y), \frac{\partial}{\partial x}(2x+3z) - \frac{\partial}{\partial y}(2y) \right)\]Evaluate to get \( abla \times \mathbf{G} = (0, 0, -4) \). For divergence:\[abla \cdot \mathbf{G} = \frac{\partial}{\partial x}(2y) + \frac{\partial}{\partial y}(2x+3z) + \frac{\partial}{\partial z}(3y) = 0 + 0 + 0 = 0\].
05

Calculate curl and divergence for (c)

For vector field \( \mathbf{H} = (x^2 - z^2, 2, 2xz) \), find:\[abla \times \mathbf{H} = \left( \frac{\partial}{\partial y}(2xz) - \frac{\partial}{\partial z}(2), \frac{\partial}{\partial z}(x^2 - z^2) - \frac{\partial}{\partial x}(2xz), \frac{\partial}{\partial x}(2) - \frac{\partial}{\partial y}(x^2 - z^2) \right)\]Substitute values to obtain \( abla \times \mathbf{H} = (0, -2z, 0) \). For divergence:\[abla \cdot \mathbf{H} = \frac{\partial}{\partial x}(x^2 - z^2) + \frac{\partial}{\partial y}(2) + \frac{\partial}{\partial z}(2xz) = 2x + 0 + 2x = 4x\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Curl of a Vector Field
The concept of the curl of a vector field is essential when exploring the rotation of vector fields in three dimensions. When you think of curl, imagine it like a measure of how much and in which way a field "twists" around a point. Mathematically, the curl of a vector field \( \mathbf{F} \) is given by \( abla \times \mathbf{F} \), where \( abla \) (nabla) is a vector differential operator.
  • A nonzero curl indicates some rotation-like behavior in the field.
  • If the curl is zero everywhere in the region, then the vector field is called irrotational.
When tackling exercise problems involving curl, it is crucial to differentiate correctly and apply the cross product. For example, for the vector field \( \mathbf{F} = (x+y, -x+y, -2z) \), the curl results as \( (0, 0, 0) \), indicating no rotation. This outcome suggests that \( \mathbf{F} \) could be a gradient field, leading us to explore potential scalar functions that \( \mathbf{F} \) might originate from.
Divergence of a Vector Field
The divergence of a vector field is a scalar value that measures the rate at which density exits or enters a given point. It reflects how much a vector field "spreads out" from or converges towards a point in space. Calculating divergence involves applying the dot product, represented as \( abla \cdot \mathbf{F} \), where \( abla \) is again the vector differential operator.
  • A positive divergence indicates a source at that point, where more vectors are emerging than entering.
  • A negative divergence indicates a sink, suggesting a net influx towards that point.
  • If the divergence is zero, the field has no net inflow or outflow at that point.
In our exercise, the vector field \( \mathbf{F} = (x+y, -x+y, -2z) \) exhibits a divergence of 0, showing neutrality in net flow at any given point. It's crucial to apply the concept of partial derivatives accurately while computing divergence to understand the behavior of the vector field at various points.
Gradient Fields
A gradient field is a vector field that can be expressed as the gradient of a scalar function, \( \mathbf{F} = abla \phi \), where \( \phi \) is the scalar potential function. These fields are conservative, meaning the work done by moving along the field from one point to another is path-independent and only depends on the endpoints.To know if a vector field is a gradient field, check if its curl is zero. This condition is necessary in simply-connected regions. Once identified, find the scalar potential by integrating each component of the vector field partially, ensuring consistency among the results.In our task, the zero curl of \( \mathbf{F} = (x+y, -x+y, -2z) \) implies it's likely a gradient field, leading us to determine \( \phi(x, y, z) = \frac{x^2}{2} - xy + \frac{y^2}{2} - z^2 \) as the potential function. Understanding this procedure helps verify whether a given vector field is conservative and explores the underlying potential functions.

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Most popular questions from this chapter

As a distribution of electric charge, the gold nucleus can be described as a sphere of radius \(6 \times 10^{-13} \mathrm{~cm}\) with a charge \(Q=79 e\) distributed fairly uniformly through its interior. What is the potential \(\phi_{0}\) at the center of the nucleus, expressed in megavolts? (First derive a general formula for \(\phi_{0}\) for a sphere of charge \(Q\) and radius \(a\). Do this by using Gauss's law to find the internal and external electric field and then integrating to find the potential.)

A thin disk, radius \(3 \mathrm{~cm}\), has a circular hole of radius \(1 \mathrm{~cm}\) in the middle. There is a uniform surface charge of \(-4 \mathrm{esu} / \mathrm{cm}^{2}\) on the disk. (a) What is the potential in statvolts at the center of the hole? (Assume zero potential at infinite distance.) (b) An electron, starting from rest at the center of the hole, moves out along the axis, experiencing no forces except repulsion by the charges on the disk. What velocity does it ultimately attain? (Electron mass \(=9 \times 10^{-28} \mathrm{gm} .\) )

Does the function \(f(x, y)=x^{2}+y^{2}\) satisfy the two-dimensional Laplace's equation? Does the function \(g(x, y)=x^{2}-y^{2}\) ? Sketch the latter function, calculate the gradient at the points \((x=\) \(0, y=1) ;(x=1, y=0) ;(x=0, y=-1) ;\) and \((x=-1, y=\) 0) and indicate by little arrows how these gradient vectors point.

An interstellar dust grain, roughly spherical with a radius of \(3 \times 10^{-7}\) meters, has acquired a negative charge such that its potential is \(-0.15\) volt. How many extra electrons has it picked up? What is the strength of the electric field at its surface, expressed in volts/ meter?

Consider a charge distribution which has the constant density \(\rho\) everywhere inside a cube of edge \(b\) and is zero everywhere outside that cube. Letting the electric potential \(\phi\) be zero at infinite distance from the cube of charge, denote by \(\phi_{0}\) the potential at the center of the cube and \(\phi_{1}\) the potential at a corner of the cube. Determine the ratio \(\phi_{0} / \phi_{1}\). The answer can be found with very little calculation by combining a dimensional argument with superposition. (Think about the potential at the center of a cube with the same charge density and with twice the edge length.)
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