Question

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

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

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 \( \phi \).

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.

Formulate the Differential of the Scalar Field

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

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 \( \hat{\phi} \).

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.

Key concepts

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.