Problem 7 If \(z=f(x, y), x=\rho \cos \the... [FREE SOLUTION] (2024)

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The chain rule in calculus is a fundamental theorem used to take derivatives of composite functions. When we have a function that is a combination of two or more functions, the chain rule enables us to differentiate it with respect to one of its variables. In the context of changing between coordinate systems, like Cartesian and polar coordinates, the chain rule becomes particularly useful.

For example, if we have a function in terms of polar coordinates, say, \( F(\rho, \theta) \), and we wish to find how \( F \) changes with respect to the variables \( x \) and \( y \) of Cartesian coordinates, we would apply the chain rule. It allows us to link the rate of change of \( F \) with respect to \( x \) and \( y \) to its rate of change with respect to \( \rho \) and \( \theta \).

By using the relationships between the coordinates, \( x = \rho \cos \theta \) and \( y = \rho \sin \theta \), we can express the partial derivatives with respect to \( \rho \) and \( \theta \) in terms of \( x \) and \( y \). This is done by differentiating the function firstly with respect to \( x \) and \( y \), and then considering the derivatives of \( x \) and \( y \) with respect to \( \rho \) and \( \theta \).

The application of the chain rule in polar coordinates, therefore, links the partial derivatives of a function in polar coordinates to its partial derivatives in Cartesian coordinates, bridging the gap between the two systems and facilitating the process of transitioning from one to another when solving problems in multi-variable calculus.

Trigonometric identities are equalities that involve trigonometric functions and are true for every value where both sides of the equality are defined. These identities are very useful when simplifying expressions and solving equations involving trigonometric functions.

Key Trigonometric Identities

Some of the most commonly used trigonometric identities include the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \), which was used in the step-by-step solution to the exercise. This identity comes from the Pythagorean theorem and the unit circle definition of sine and cosine functions.
Another set of useful identities is the angle sum and difference identities, such as \( \sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta) \) and \( \cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta) \).

When dealing with polar coordinates, these identities allow us to connect the trigonometric functions of \( \theta \) with the Cartesian coordinates, leading to simplifications that can be vital for solving calculus problems. Understanding and utilizing these identities are crucial for students working with trigonometric functions and is especially important in the study of calculus, where they frequently appear in integration and differentiation problems.

Transformation between coordinate systems is a common task in mathematics, physics, and engineering. The polar to Cartesian coordinate transformation is one such example, wherein we switch from the polar coordinate system (\( \rho, \theta \)) to the Cartesian coordinate system (\( x, y \)). This process often utilizes the relationships \( x = \rho \cos \theta \) and \( y = \rho \sin \theta \), derived from the definition of polar coordinates.

Converting Between Systems

The conversion formulas instantiate how a point's position is viewed in different systems. In the polar system, a point's location is identified by the distance from the origin, \( \rho \), and the angle, \( \theta \), from the positive x-axis. In the Cartesian system, the point's position is given by its horizontal (x) and vertical (y) distances from the origin.

These transformations are not only critical for changing the coordinate representation of points but also play a significant role in problems involving calculus, as seen in the exercise provided. The ability to convert between polar and Cartesian coordinates allows one to select the most convenient framework for tackling a problem, which can lead to simpler computations or more intuitive solutions. For calculus students, mastering these transformations is essential when working with functions represented in different coordinates and when integrating or differentiating in multiple dimensions.