Question

Recall that Cartesian and polar coordinate are related through the transformation equations shown on the right. a. Evaluate the partial derivatives Xr, Yr, X θ and y θ.

Answer 1

The partial derivatives of Cartesian coordinates x and y with respect to polar coordinates r and θ are cos(θ) and sin(θ) for Xr and Yr respectively, and -r × sin(θ) and r × cos(θ) for Xθ and Yθ respectively.

Step-by-step explanation:

The given problem involves calculating the partial derivatives of Cartesian coordinates (x, y) with respect to polar coordinates (r, θ). The transformations between Cartesian and polar coordinates are given by:

• x = r × cos(θ)
• y = r × sin(θ)

To find the partial derivatives:

• The partial derivative of x with respect to r, denoted as Xr, is given by d(x)/d(r) = cos(θ).
• The partial derivative of y with respect to r, denoted as Yr, is given by d(y)/d(r) = sin(θ).
• The partial derivative of x with respect to θ, denoted as Xθ, is given by d(x)/d(θ) = -r × sin(θ).
• The partial derivative of y with respect to θ, denoted as Yθ, is given by d(y)/d(θ) = r × cos(θ).

These partial derivatives describe how the Cartesian coordinates x and y change with respect to changes in the polar coordinates r and θ.