How Do Trigonometric Identities Derive the Area Element in Polar Coordinates?

[markapplegate]

Homework Statement

using only trigonometric identities, derive the differential area element in polar coordinates? any help with this problem or at least a start?

Homework Equations

i found this so far
dA=(dr)(rd θ)

The Attempt at a Solution

i have tried to figure this one out but i really have no clue how to start the problem, i tried taking derivatives but got no where , I am not sure how to use a trigonometric identitie in this problem?

 

[lanedance]

geomtrically:

say you have a vector given by r = (r, θ)

think of a small "square" at the end of the vector,
one side length is given by moving in the r dierction by dr
the other side is swept by changing θ by dθ, the length is a circular arc so will be r.dθ

so the "square" area elemnt is givne by
dA = r.dθ.dr

algebraically:

you know for cartesian coordinates dA = dx.dy
write x & y in terms of θ & r, then take the partial derivatives to find dx(r,θ)

 

[markapplegate]

thank you, i haven't learned partial derivatives yet, but i will try to figure that out and I am guessing i will be able to use some trig id. after i take the partial derivative.