Chapter 4: Q19P (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.
$\frac{\partial^{2} z}{\partial r \partial y}$

Short Answer

Expert verified

The value of provided equation is $\frac{\partial^{2} z}{\partial r \partial y} = 0$ .

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

From equation 1 and 2,

$z = {(r \cos θ)}^{2} + 2 y^{2} z = r^{2} \cos^{2} θ + 2 y^{2}$

Take the partial derivative of z with respect to r,

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

Take the partial derivative of $\frac{\partial z}{\partial r}$ with respect to y,

$\frac{\partial^{2} z}{\partial z \partial y} = \frac{\partial^{2}}{\partial z \partial y} 2 r \cos^{2} θ \frac{\partial^{2} z}{\partial z \partial y} = 0$

Hence, $\frac{\partial^{2} z}{\partial z \partial y} = 0$

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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.
$\frac{\partial^{2} z}{\partial r \partial y}$

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.∂2z∂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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Study anywhere. Anytime. Across all devices.

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