   Chapter 12.5, Problem 71E

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# Curvature A curve C is given by the polar equation r = f ( θ ) . Show that the curvature K at the point ( r , θ ) is K = | 2 ( r ' ) 2 − r r " + r 2 | [ ( r ' ) 2 + r 2 ] 3 / 2 [Hint: Represent the curve by r ( θ ) = r cos θ i + r sin θ j .]

To determine

To prove: That the curvature k for the polar equation at the point (r, θ) is

k=|2(r)2r.r+r2|[(r)2+r2]32.

Explanation

Given:

The function is: r=f(θ).

Formula used:

Derivative product rule.

ddx(uv)=uv+uv

Proof:

The basic equation of the curvature is given as:

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

Now, x(θ)=f(θ)cosθ  and y(θ)=f(θ)sinθ.

x(θ)=f(θ)cosθ+f(θ)sinθx(θ)=f(θ)cosθf(θ)sinθf(θ)sinθ+f(θ)cosθy(θ)=f(θ)sinθ+f(θ)cosθ+f(θ)+f(θ)sinθ

Now,

(xyyx)=((f(θ)sinθ+f(θ)cosθ)  (f(θ)sinθ+2f(θ)cosθ+f(θ)sinθ) (f(θ)cosθ+f(θ)sinθ) (f(θ)cosθ2f(θ)sinθ+f(θ)cosθ))={f2(θ)sin2θf(θ)f(θ)sinθcosθ2f(θ)f(θ)sin<

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