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Chapter 4: Q22P (page 191)

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

鈭�2z鈭�r鈭�x.

Short Answer

Expert verified

The value of provided equation is 鈭�2z鈭�r鈭�x=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

Square the equation 2 and 3, then add.

x2+y2=r2 ...(4)

From equation (1),

z=x2+y2+y2z=r2+x2tan2胃

Take the partial derivative of z with respect to ,

鈭�z鈭�r=鈭�鈭�rr2+x2tan2胃鈭�z鈭�r=2r

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

role="math" localid="1659090151351" 鈭�2z鈭�r鈭�x=鈭�2鈭�r鈭�x2r鈭�2z鈭�r鈭�x=0

Hence,

鈭�2z鈭�r鈭�x=0

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Most popular questions from this chapter

For w=8x4+y4-2xy2, find鈭�2w/鈭�x2and鈭�2w/鈭�y2at the points where鈭�w/鈭�x=鈭�w/鈭�y=0.

Question: Find the shortest distance from the origin to each of the following quadric surfaces. Hint: See Example 3 above.

4y2+2z2+3x=18

To find the maximum and the minimum points of the given function.

x2+y2+2x-4y+10

(a). Given the point (2,1)in the(x,y) plane and the line 3x+2y=4, find the distance from the point to the line by using the method of Chapter 3, Section 5.

(b). Solve part (a) by writing a formula for the distance from (2,1)to(x,y) and minimizing the distance (use Lagrange multipliers).

(c). Derive the formula

D=|ax0+by0-ca2+b2|

For the distance from (x0,y0)to ax+by=cby the methods suggested in parts (a) and (b).

Let Rbe the resistance ofR1=25ohmsandR2=15ohmsohms in parallel. (See Chapter 2, Problem 16.6.) IfR1is changed to25.1ohms, findR2so thatRis not changed.
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