91Ó°ÊÓ

Partial derivatives are essential when dealing with functions of multiple variables. They measure how a function changes as one of the input variables changes, while the others are held constant. In this problem, the function in question is \( w = f(x, y) \), where we substitute the Cartesian coordinates \((x, y)\) with polar coordinates \((r, \theta)\). This requires us to compute the partial derivative of \( w \) first with respect to \( r \) and then with respect to \( \theta \).

• \( \frac{\partial w}{\partial r} \) represents the rate of change of \( w \) as \( r \) changes, with angle \( \theta \) fixed.
• \( \frac{\partial w}{\partial \theta} \) represents the rate of change of \( w \) as \( \theta \) changes, with radius \( r \) fixed.

During this substitution, the key is to express the changes in \( w \) not directly through \( x \) and \( y \), but through their equivalents in polar terms. Understanding these expressions and their derivation helps in visualizing how a multivariable function behaves in polar coordinates.

The chain rule is a fundamental concept in calculus used to compute the derivative of a composite function. When converting Cartesian coordinates to polar coordinates in this exercise, the chain rule helps express derivatives in terms of \( r \) and \( \theta \). It essentially "chains" together the derivatives of inner functions to determine the overall rate of change.

Here's how the chain rule works in our problem:

• For \( \frac{\partial w}{\partial r} \), the chain rule combines \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \) with the derivatives of the polar expressions for \( x \) and \( y \) with respect to \( r \) (i.e., \( \cos \theta \) and \( \sin \theta \)).
• For \( \frac{\partial w}{\partial \theta} \), it involves multiplying the partial derivatives of \( f \) by the derivatives \( -r \sin \theta \) and \( r \cos \theta \) — derivatives of \( x \) and \( y \) with respect to \( \theta \).

Understanding the chain rule is vital here because it allows you to bridge the calculations from one coordinate system to another without losing essential information about the function's behavior.

Polar formulas represent relationships that arise when converting between Cartesian and polar coordinates. These are crucial when working in contexts involving circular or rotational symmetries, where expressing problems in terms of \( r \) and \( \theta \) simplifies the computations.

In this exercise, we particularly utilize the conversions:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( \frac{\partial x}{\partial r} = \cos \theta, \frac{\partial y}{\partial r} = \sin \theta \)
• \( \frac{\partial x}{\partial \theta} = -r \sin \theta, \frac{\partial y}{\partial \theta} = r \cos \theta \)

These formulas showcase how changes in \( r \) and \( \theta \) affect \( x \) and \( y \), capturing the dynamics of transformations between the two systems.

Finally, the relationship \((f_x)^2 + (f_y)^2 = \left(\frac{\partial w}{\partial r}\right)^2 + \frac{1}{r^2}\left(\frac{\partial w}{\partial \theta}\right)^2\) consolidates how polar derivatives relate to the magnitudes of the corresponding partial derivatives \( f_x \) and \( f_y \), making it easier to compare and analyze them.