Question

Assume that all the given functions have continuous second-order partial derivatives. If z = f(x, y), where x = 9r cos(θ) and y = 9r sin(θ), find the following: A) ∂z / ∂r B) ∂z / ∂θ C) ∂^2z / ∂r ∂θ

Answer 1

Answer with Step-by-step explanation:

We are given that all the given functions have continuous second-order partial derivatives.

`z=f(x,y)`

Where
`x=9rcos\theta,y=9r sin\theta`

We have to find

A.
`(\delta z)/(\delta r)`

We know that

`(\delta z)/(\delta r)=(\delta z)/(\delta x)(\delta x)/(\delta r)+(\delta z)/(\delta y)(\delta y)/(\delta r)`

Using this formula

`(\delta z)/(\delta r)=9cos\theta (\delta z)/(\delta r)+9sin\theta(\delta z)/(\delta r)`

`(\delta z)/(\delta r)=(x)/(r)(\delta z)/(\delta r)+(y)/(r)\frac{\delta z}\delta y}`

B.
`(\delta z)/(\delta \theta)`

`(\delta z)/(\delta \theta)=(\delta z)/(\delta x)(\delta x)/(\delta\theta )+(\delta z)/(\delta y)(\delta y)/(\delta\theta)`

`(\delta z)/(\delta \theta)=-9rsin\theta(\delta z)/(\delta x)+9rcost\theta(\delta z)/(\delta y)`

`(\delta z)/(\delta \theta)=-y(\delta z)/(\delta x)+x(\delta z)/(\delta y)`

C.
`(\delta^2 z)/(\delta r\delta\theta)`

`(\delta^2 z)/(\delta r\delta\theta)=-9sin\theta(\delta z)/(\delta x)-y(\delta^2z)/(\delta x^2)(9cos\theta)+9 cos\theta(\delta z)/(\delta y)+x(\delta^2z)/(\delta y^2)(9sin\theta)`

`(\delta^2 z)/(\delta r\delta\theta)=-9sin\theta(\delta z)/(\delta x)-(81rsin\theta cos\theta)(\delta^2z)/(dx^2)+9cos\theta(\delta z)/(\delta y)+(81r cos\theta sin\theta)(\delta^2z)/(\delta y^2)`

`(\delta^2 z)/(\delta r\delta\theta)=-(y)/(r)(\delta z)/(\delta x)-(xy)/(r)(\delta^2)/(\delta x^2)+(x)/(r)(\delta z)/(\delta y)+(xy)/(r)(\delta^2z)/(\delta y^2)`