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Chapter 12: Q 49. (page 944)

In Exercises 43-50, compute all of the second-order partial derivatives for the functions fr,θ=rsinθand show that the mixed partial derivatives are equal.

Short Answer

Expert verified

The second order partial derivatives for the function are

∂2f∂r2=0∂2f∂θ2=-rsinθ

Step by step solution

01

Step 1. Definition 

Clairaut's theorem on equality of mixed partials states that under assumption of continuity of both the second-order mixed partials of a function of two variables, the two mixed partials are equal.

02

Step 2. Finding second order partial derivative 

fr,θ=rsinθ⋯⋯1
Partially differentiate equation 1both sides with respect tor

⇒∂f∂r=sinθ

Again partially differentiate both sides with respect tor

⇒∂2f∂r2=0

Partially differentiate equation 1both sides with respect toθ

⇒∂f∂θ=rcosθ

Again partially differentiate both sides with respect to θ

⇒∂2f∂θ2=-rsinθ

03

Step 3. Finding mixed order partial derivative  

fr,θ=rsinθ

⇒∂2f∂r∂θ=∂∂rrcosθ⇒∂2f∂r∂θ=cosθ

Also

⇒∂2f∂θ∂r=∂∂θsinθ⇒∂2f∂θ∂r=cosθ

Now as we observe

∂2f∂r∂θ=∂2f∂θ∂r[Hence proved]

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Most popular questions from this chapter

Extrema:Findthelocalmaxima,localminima,andsaddlepointsofthegivenfunctionsf(x,y)=x3+y3−6x2+3y2-4.

Construct examples of the thing(s) described in

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