Chapter 4: Q11P (page 191)

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

Short Answer

Expert verified

Hence the value of provide equation is ${(\frac{\partial z}{\partial y})}_{r} = 2 y$

Step by step solution

Explanation of solutionStep 1: 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, and then add.

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

From equation (1),

$\begin{array}{l} z = (x^{2} + y^{2}) + y^{2} \\ z = r^{2} + y^{2} \end{array}$

Take the partial derivative of z with respect to y,

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

Hence,

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

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

Short Answer

Step by step solution

Explanation of solutionStep 1: Given data

Partial differentiation

Calculation

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

Short Answer

Step by step solution

Explanation of solutionStep 1: Given data

Partial differentiation

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

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