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Find the components of \(\nabla f(r, \phi)\) in two-dimensional polar coordinates. [Hint: Remember that the change in the scalar \(f \text { as a result of an infinitesimal displacement } d \mathbf{r} \text { is } d f=\nabla f \cdot d \mathbf{r}.]\)

Short Answer

Expert verified
Components are \( \frac{\partial f}{\partial r} \hat{r} \) and \( \frac{1}{r} \frac{\partial f}{\partial \phi} \hat{\phi} \) in polar coordinates.

Step by step solution

01

Recall the Gradient in Polar Coordinates

The gradient of a scalar field \( f(r, \phi) \) in polar coordinates is given by the vector \( abla f \) such that \( df = abla f \cdot d \mathbf{r} \). Here, \( d\mathbf{r} \) is expressed in terms of infinitesimal changes in \( r \) and \( \phi \).
02

Express Infinitesimal Displacement

In polar coordinates, an infinitesimal displacement \( d \mathbf{r} \) can be expressed as \( d\mathbf{r} = dr \hat{r} + r d\phi \hat{\phi} \), where \( \hat{r} \) and \( \hat{\phi} \) are unit vectors in the directions of increasing \( r \) and \( \phi \), respectively.
03

Formulate the Differential of the Scalar Field

The change in the scalar field \( f(r, \phi) \) due to this displacement is written as \( df = \frac{\partial f}{\partial r} dr + \frac{\partial f}{\partial \phi} d\phi \).
04

Interpret the Dot Product

From \( df = abla f \cdot d \mathbf{r} \), the expression becomes: \[ df = \left( \frac{\partial f}{\partial r} \right) dr + \left( \frac{1}{r} \frac{\partial f}{\partial \phi} \right) r d\phi. \] This implies that the unit vectors' contribution to the gradient is split between the scalar \( \frac{\partial f}{\partial r} \) in the direction of \( \hat{r} \) and \( \frac{1}{r} \frac{\partial f}{\partial \phi} \) in the direction of \( \hat{\phi} \).
05

Determine the Components of \( \nabla f \)

Thus, the components of the gradient \( abla f \) in polar coordinates are \( abla f = \frac{\partial f}{\partial r} \hat{r} + \frac{1}{r} \frac{\partial f}{\partial \phi} \hat{\phi} \). This indicates how the function \( f(r, \phi) \) changes in both radial and angular directions.

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Key Concepts

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

Partial Derivatives
Partial derivatives are a fundamental concept in multivariable calculus, representing how a function changes as one of its input variables is varied, while keeping other variables constant. In the context of polar coordinates, when we have a function \( f(r, \phi) \), its partial derivatives are \( \frac{\partial f}{\partial r} \) and \( \frac{\partial f}{\partial \phi} \). These notations signify the rate of change of the function with respect to \( r \) and \( \phi \), respectively.
When dealing with polar coordinates, partial derivatives must be carefully evaluated because changes in \( r \) and \( \phi \) can affect the function differently compared to Cartesian coordinates. Below are some key points about partial derivatives:
  • The partial derivative \( \frac{\partial f}{\partial r} \) measures how much the function changes as we move radially outward from the origin.
  • The derivative \( \frac{\partial f}{\partial \phi} \) measures the rate of change as the angle \( \phi \) changes.
  • In polar coordinates, these derivatives help build a more nuanced picture about how functions change across a plane.
Understanding partial derivatives helps us determine and analyze the gradient of a function, especially in polar coordinates.
Polar Coordinates
Polar coordinates offer a different way of representing points in a plane from the traditional Cartesian coordinates. Instead of using \( (x, y) \), polar coordinates use \( (r, \phi) \), where \( r \) is the radial distance from the origin, and \( \phi \) is the angular position measured from the positive x-axis.
This system can simplify the equations of curves and make it easier to handle rotation and scaling. Several aspects of polar coordinates are crucial:
  • Polar coordinates are particularly helpful for describing regions and shapes that are naturally circular or involve rotation.
  • The relationship between Cartesian and polar coordinates is given by the transformations: \( x = r\cos\phi \) and \( y = r\sin\phi \).
  • In polar coordinates, the orientation is naturally radial, with angles leading to different expansions opposed to Cartesian which is very linear.
The switch to polar coordinates necessitates a change in how we handle calculus operations, leading to adjusted definitions like those of the gradient.
Scalar Field
A scalar field is a mathematical function that assigns a single scalar value to every point in space. Imagine it as a landscape where each point has a height, corresponding to the scalar value. In the context of our exercise, this scalar field is \( f(r, \phi) \).
Scalar fields can represent various physical quantities, such as:
  • Temperature distribution in a room.
  • Pressure in a gas.
  • Gravitational potential in space.
Our task is to understand how the scalar field, defined by \( f(r, \phi) \), changes over small movements through space. To achieve this understanding, we use gradients, which measure these changes as vectors involving the use of partial derivatives achieved in each coordinate system. The presence of a scalar field makes it possible to explore these spatial variations effectively using calculus and coordinate transformations.

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Most popular questions from this chapter

(a) Write down the Lagrangian for a particle moving in three dimensions under the influence of a conservative central force with potential energy \(U(r),\) using spherical polar coordinates \((r, \theta, \phi)\). (b) Write down the three Lagrange equations and explain their significance in terms of radial acceleration, angular momentum, and so forth. (The \(\theta\) equation is the tricky one, since you will find it implies that the \(\phi\) component of \(\ell\) varies with time, which seems to contradict conservation of angular momentum. Remember, however, that \(\ell_{\phi}\) is the component of \(\ell\) in a variable direction.) (c) Suppose that initially the motion is in the equatorial plane (that is, \(\theta_{0}=\pi / 2\) and \(\dot{\theta}_{0}=0\) ). Describe the subsequent motion. (d) Suppose instead that the initial motion is along a line of longitude (that is, \(\dot{\phi}_{0}=0\) ). Describe the subsequent motion.

Consider a mass \(m\) moving in two dimensions with potential energy \(U(x, y)=\frac{1}{2} k r^{2},\) where \(r^{2}=x^{2}+y^{2} .\) Write down the Lagrangian, using coordinates \(x\) and \(y,\) and find the two Lagrange equations of motion. Describe their solutions. [This is the potential energy of an ion in an "ion trap," which can be used to study the properties of individual atomic ions.]

Consider a particle of mass \(m\) and charge \(q\) moving in a uniform constant magnetic field \(\mathbf{B}\) in the \(z\) direction. (a) Prove that \(\mathbf{B}\) can be written as \(\mathbf{B}=\nabla \times \mathbf{A}\) with \(\mathbf{A}=\frac{1}{2} \mathbf{B} \times \mathbf{r} .\) Prove equivalently that in cylindrical polar coordinates, \(\mathbf{A}=\frac{1}{2} B \rho \hat{\phi}\). (b) Write the Lagrangian (7.103) in cylindrical polar coordinates and find the three corresponding Lagrange equations. (c) Describe in detail those solutions of the Lagrange equations in which \(\rho\) is a constant.

Lagrange's equations in the form discussed in this chapter hold only if the forces (at least the nonconstraint forces) are derivable from a potential energy. To get an idea how they can be modified to include forces like friction, consider the following: A single particle in one dimension is subject to various conservative forces (net conservative force \(=F=-\partial U / \partial x)\) and a nonconservative force (let's call it \(F_{\text {fric }}\) ). Define the Lagrangian as \(\mathcal{L}=T-U\) and show that the appropriate modification is $$\frac{\partial \mathcal{L}}{\partial x}+F_{\mathrm{fric}}=\frac{d}{d t} \frac{\partial \mathcal{L}}{\partial \dot{x}}.$$

A particle is confined to move on the surface of a circular cone with its axis on the \(z\) axis, vertex at the origin (pointing down), and half-angle \(\alpha\). The particle's position can be specified by two generalized coordinates, which you can choose to be the coordinates \((\rho, \phi)\) of cylindrical polar coordinates. Write down the equations that give the three Cartesian coordinates of the particle in terms of the generalized coordinates ( \(\rho, \phi\) ) and vice versa.

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