Question

Explain why the element of area in Cartesian coordinates \(d x d y\) becomes \(r d r d \theta\) in polar coordinates.

Short answer

Based on the above solution, the relationship between the Cartesian area element (dx dy) and the polar area element (r dr dθ) is given by the equation: dx dy = r dr dθ. This means that in polar coordinates, the area element is a product of the radial distance (r) and the infinitesimal changes in the radial (dr) and angular (dθ) variables. The difference between the two is that the Cartesian area element is obtained by multiplying infinitesimal changes in the x and y directions, whereas in polar coordinates, the area element is based on radial distance and small changes in radial and angular components.

Step-by-step solution

Use the coordinate transformation

In order to establish a relationship between the different area elements, we'll first write the basic coordinate transformation from Cartesian to polar coordinates: \((x, y) = (r \cos(\theta), r \sin(\theta))\) Note that for small changes in r and θ, this transformation becomes: \((\Delta x, \Delta y) = (\Delta r \cos(\theta), \Delta r \sin(\theta))\) If we replace Δr with dr and Δθ with dθ, we have: \((\Delta x, \Delta y) = (dr \cos(\theta), dr \sin(\theta))\)

Find the magnitudes of the coordinate transformation

Next, we'll square both sides of the above transformation and add them up to find the magnitudes: \((\Delta x)^2 + (\Delta y)^2 = (dr \cos(\theta))^2 + (dr \sin(\theta))^2\) Upon simplification, we have: \(\Delta x^2 + \Delta y^2 = dr^2 \cos^2(\theta) + dr^2 \sin^2(\theta) = dr^2(\cos^2(\theta) + \sin^2(\theta))\) Since \(\cos^2(\theta) + \sin^2(\theta) = 1\), we get: \(\Delta x^2 + \Delta y^2 = dr^2\) Now, let's consider Δx and Δy separately and find their components in terms of r and θ: \(\Delta x = dr \cos(\theta)\) \(\Delta y = dr \sin(\theta)\)

Compute the area element in Cartesian and polar coordinates

We have, in Cartesian coordinates, the area element dA is: \(dA = dx \ dy\) Now, in polar coordinates, the area element is given by: \(dA = r~d\theta \ dr\)

Show the relationship between area elements

We'll now calculate the product of the small changes dx and dy in Cartesian coordinates using the transformed components. Multiply Δx and Δy together to have: \(\Delta x \Delta y = dr \cos(\theta) \cdot dr \sin(\theta) = r(\cos(\theta) \sin(\theta)) dr~d\theta\) Now, for infinitesimally small elements, we can write this as: \(dx dy = r \cos(\theta) \sin(\theta) dr~d\theta\) As \(\cos(\theta) \sin(\theta)\) is the Jacobian of the coordinate transformation, the last equation can be rewritten as: \(dx dy = r \cdot dr~d\theta\) This shows that the Cartesian area element \(dx dy\) is replaced by the polar area element \(r dr~d\theta\) in polar coordinates, as required.