Chapter 5: Double Integration
Section 5.7: Double Integration in Polar Coordinates
Give a geometric construction showing that for polar coordinates, dA′=r dr dθ or r dθ dr.
In the Cartesian plane, the area of a rectangle formed by the vectors Rx=dx i and Ry=dy j is Rx×Ry=dx dy, where to compute the cross product, zeros are added as a third component in each vector.
If R=r cos(θ)r sin(θ)0 is the position vector under a transformation to polar coordinates, vectors tangent to the edges of a transformed rectangle would be given by the vectors
∂R∂r dr=cos(θ)sin(θ)0 dr and ∂R∂θdθ=−r sin(θ)r cos(θ)0 dθ
The area of the deformed rectangle, nearly a parallelogram for dr and dθ "small," is then
∂R∂r dr×∂R∂θ dθ = 0 i+0 j+r dr dθ k = r dr dθ
Maple Solution - Interactive
Tools≻Load Package: Student Multivariate Calculus
Context Panel: Assign Name
R=r cosθ,r sinθ,0→assign
Obtain ∂R∂r dr×∂R∂θ dθ
Calculus palette: Partial-differentiation operator
Context Panel: Evaluate and Display Inline
Context Panel: Simplify≻Assuming Positive
∂∂ r R dr×∂∂ θ R dθ = dr⁢cos⁡θ2⁢dθ⁢r+dr⁢sin⁡θ2⁢dθ⁢r2→assuming positivedr⁢r⁢dθ
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