   Chapter 12.5, Problem 74E

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# Curvature at the Pole Show that the formula for the curvature of a polar curve r = f ( θ ) given in Exercise 69 reduces to K = 2 / | r ' | for the curvature at the pole.

To determine

To prove: The formula for the curvature of a polar curve reduces to K=2|r| for the curvature at the pole.

Explanation

Given:

The function r=f(θ)

Formula used

Curvature K=[xyyx][(x)2+(y)2]3/2

Proof:

Consider the general equation of r=f(θ).

Since r(θ)=rcosθi+rsinθj, the function r(θ) can be expressed as follows:

r(θ)=rcosθi+rsinθj=f(θ)cosθi+f(θ)sinθj

So,

x(θ)=f(θ)cosθ,y(θ)=f(θ)sinθ

Differentiate above obtained equations with respect to θ as follows:

x(θ)=f(θ)sinθ+f(θ)cosθy(θ)=f(θ)cosθ+f(θ)sinθ

Differentiate above differentiated equations with respect to θ again to obtain:

x(θ)=f(θ)cosθf(θ)sinθf(θ)sinθ+f(θ)cosθ=f(θ)cosθ2f(θ)sinθ+f(θ)cosθ

And,

y(θ)=f(θ)sinθ+f(θ)cosθ+f(θ)cosθ+f(θ)sinθ=f(θ)

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