Given the polar coordinates x = rcosθ and y = rsinθ, we need to find the differential area element, dxdy.
In polar coordinates, the differential elements dx and dy are derived based on the definitions of x and y.
For x = rcosθ, differentiating both sides gives us dx = -rsinθ dθ + dr cosθ.
For y = rsinθ, differentiating both sides gives us dy = rcosθ dθ + dr sinθ.
The differential area element in polar coordinates, dxdy, is obtained by multiplying the expressions of dx and dy together:
dxdy = dx * dy
= (-rsinθ dθ + dr cosθ) * (rcosθ dθ + dr sinθ)
Applying the product rule here and simplifying, we get:
dxdy = dr² cosθ sinθ + r² cosθ sinθ dθ² - dr * r* sin^2θ dθ - dr*r*cos^2θ dθ.
We then rearrange the terms and combine them to get the final form of dxdy = r * dr * dθ.
Thus, in polar coordinates, the differential area element is r * dr * dθ. Keep in mind that this is a symbolic calculation and can't be computed until actual numeric values for r, dr, and dθ are given.