Chapter 4: Q17P (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.
${(\frac{\partial z}{\partial r})}_{x}$

Short Answer

Expert verified

The value of provided equation is $(\frac{\partial z}{\partial r}), = 4 r$ .

Step by step solution

Given data

The given data are listed below

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

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),

$z = x^{2} + 2 (x^{2} + y^{2}) - 2 x^{2} z = - x^{2} + 2 r^{2}$

Now, differentiate partially with respect to ,

${(\frac{\partial z}{\partial r})}_{x} = \frac{\partial}{\partial r} [- x^{2} + 2 r^{2}] {(\frac{\partial z}{\partial r})}_{x} = 4 r$

Hence, ${(\frac{\partial z}{\partial r})}_{x} = 4 r$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

Short Answer

Step by step solution

Given data

Partial differentiation

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Circular Motion and Gravitation

Nuclear Physics

Particle Model of Matter

Wave Optics

Famous Physicists

Study anywhere. Anytime. Across all devices.

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

Short Answer

Step by step solution

Given data

Partial differentiation

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Circular Motion and Gravitation

Nuclear Physics

Particle Model of Matter

Wave Optics

Famous Physicists

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.
${(\frac{\partial z}{\partial r})}_{x}$