Chapter 4: Q18P (page 191)

If, $z = x^{2} + 2 y^{2}, x = r c o s θ, y = r s i n θ$ , find the following partial derivatives.
${(\frac{\partial z}{\partial r})}_{y}$

Short Answer

Expert verified

The value of provided equation is ${(\frac{\partial z}{\partial r})}_{y} = 2 r$ .

Step by step solution

Given data

The given data are listed below

$z = x^{2} + 2 y^{2} . . . (1) x = r \cos θ . . . . . (2) y = r \sin θ . . . . (3)$

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.

Calculation

Square the equation 2 and 3, then add.

$x^{2} + y^{2} = r^{2}$ ...(4)

From equation (1),

$_{z = r^{2} + y^{2}}^{z = [x^{2} + y^{2}] + y^{2}} z = x^{2} + 2 y^{2}$

Take the partial derivative of z with respect to ,

${(\frac{\partial z}{\partial r})}_{y} = \frac{\partial}{\partial r} (r^{2} + y^{2}) {(\frac{\partial z}{\partial r})}_{y} = 2 r$

Hence, ${(\frac{\partial z}{\partial r})}_{y} = 2 r$

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If, $z = x^{2} + 2 y^{2}, x = r c o s θ, y = r s i n θ$ , find the following partial derivatives.
${(\frac{\partial z}{\partial r})}_{y}$

Short Answer

Step by step solution

Given data

Partial differentiation

Calculation

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If,z=x2+2y2,x=rcosθ,y=rsinθ, find the following partial derivatives.(∂z∂r)y

Short Answer

Step by step solution

Given data

Partial differentiation

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

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Linear Momentum

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If, $z = x^{2} + 2 y^{2}, x = r c o s θ, y = r s i n θ$ , find the following partial derivatives.
${(\frac{\partial z}{\partial r})}_{y}$