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Chapter 4: Q19P (page 191)

If,z=x2+2y2,x=rcos胃,y=rsin胃, find the following partial derivatives.

鈭�2z鈭�r鈭�y

Short Answer

Expert verified

The value of provided equation is 鈭�2z鈭�r鈭�y=0.

Step by step solution

01

Given data

The given data are listed below

z=x2+2y2....(1)x=rcos胃.....(2)y=rsin胃......(3)

02

Partial differentiation

The procedure of calculating the partial derivative of a function is called partial differentiation. This method is used to get the partial derivative of a function with respect to one variable while keeping the other constant.

03

Calculation

From equation 1 and 2,

z=rcos胃2+2y2z=r2cos2胃+2y2

Take the partial derivative of z with respect to r,

鈭�z鈭�r=鈭�鈭�r(r2cos2胃+2y2)鈭�z鈭�r=2rcos2胃

Take the partial derivative of 鈭�z鈭�rwith respect to y,

鈭�2z鈭�z鈭�y=鈭�2鈭�z鈭�y2rcos2胃鈭�2z鈭�z鈭�y=0

Hence,鈭�2z鈭�z鈭�y=0

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if z=x2+2y2,x=rcos胃,y=sin胃, find the following partial derivatives.

(鈭�z鈭�x)胃.
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