Chapter 4: Q15P (page 191)

If $z = x^{2} + 2 y^{2}, x = r \cos θ, y = r \sin θ,$ find the following partial derivatives.
${(\frac{\partial z}{\partial θ})}_{r}$ .

Short Answer

Expert verified

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

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, 2 and 3,

$\begin{array}{l} z = {(r \cos θ)}^{2} + 2 (r \sin θ) \end{array}$ ²z=r2(cos2θ+2sin2θ)

Take the partial derivative of with respect to ' $θ$ ',

$\frac{\partial z}{\partial θ} = \frac{\partial}{\partial θ} [r^{2} (1 + \sin^{2} θ)] = 2 r^{2} \sin θ \cos θ$

Now, differentiate partially with respect to 'r'

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If $z = x^{2} + 2 y^{2}, x = r \cos θ, y = r \sin θ,$ find the following partial derivatives.
${(\frac{\partial z}{\partial θ})}_{r}$ .

Short Answer

Step by step solution

Given data

Partial differentiation

Calculation

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

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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If $z = x^{2} + 2 y^{2}, x = r \cos θ, y = r \sin θ,$ find the following partial derivatives.
${(\frac{\partial z}{\partial θ})}_{r}$ .