Q5MP Given z=y2聽- 2x2, Find (鈭倆鈭�... [FREE SOLUTION]

Chapter 4: Q5MP (page 238)

Given $z = y^{2}$ - $2 x^{2}$ , Find ${(\frac{鈭� z}{鈭� x})}_{r}, {(\frac{鈭� z}{鈭� 胃})}_{x}, \frac{{鈭�}^{2} z}{鈭� x 鈭� 胃}$ .

Short Answer

Expert verified

The value of ${(\frac{鈭� z}{鈭� x})}_{r} = - 6 x, {(\frac{鈭� z}{鈭� 胃})}_{x} = 2 x^{2} 迟补苍胃 . \sec^{2} 胃 \frac{{鈭�}^{2} z}{鈭� x 鈭� 胃} = 4 x 迟补苍胃 . \sec^{2} 胃$ .

Step by step solution

Given Information

Given that $z = y^{2} - 2 x^{2}$ and the polar coordinates are $x = r \cos 胃, y = r \sin 胃$

Formula Used

The value of $r^{2} = x^{2} + y^{2}$ .

Finding the values of (∂z∂x)r,(∂z∂θ)x,∂2z∂x∂θ

Using the formula,

$\frac{鈭� z}{鈭� x} = - 6 x {(\frac{鈭� z}{鈭� x})}_{r} = - 6 x$

We know that $\tan 胃 = \frac{y}{x} 鈬� y = x \tan 胃$

Then

$z = x^{2} \tan^{2} 胃 - 2 x^{2} \frac{鈭� z}{鈭� 胃} = 2 x^{2} \tan^{2} 胃 - 2 x^{2} {(\frac{鈭� z}{鈭� 胃})}_{x} = 2 x^{2} \tan 胃 . s e c^{2} 胃$

Solving further

$\frac{{鈭�}^{2} z}{鈭� x 鈭� 胃} = \frac{鈭�}{鈭� x} (\frac{鈭� z}{鈭� 胃}) \frac{{鈭�}^{2} z}{鈭� x 鈭� 胃} = \frac{鈭�}{鈭� x} (2 x^{2} \tan 胃 . s e c^{2} 胃) \frac{{鈭�}^{2} z}{鈭� x 鈭� 胃} = 4 x \tan 胃 . s e c^{2} 胃$

Hence ${(\frac{鈭� z}{鈭� x})}_{r} = - 6 x, {(\frac{鈭� z}{鈭� 胃})}_{x} = 2 x^{2} 迟补苍胃 . \sec^{2} 胃 \frac{{鈭�}^{2} z}{鈭� x 鈭� 胃} = 4 x 迟补苍胃 . \sec^{2} 胃$ .

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91影视

Given $z = y^{2}$ - $2 x^{2}$ , Find ${(\frac{鈭� z}{鈭� x})}_{r}, {(\frac{鈭� z}{鈭� 胃})}_{x}, \frac{{鈭�}^{2} z}{鈭� x 鈭� 胃}$ .

Short Answer

Step by step solution

Given Information

Formula Used

Finding the values of (∂z∂x)r,(∂z∂θ)x,∂2z∂x∂θ

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91影视

Given z=y2- 2x2, Find (鈭�z鈭�x)r,(鈭�z鈭�胃)x,鈭�2z鈭�x鈭�胃.

Short Answer

Step by step solution

Given Information

Formula Used

Finding the values of (∂z∂x)r,(∂z∂θ)x,∂2z∂x∂θ

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Scientific Method Physics

Measurements

Waves Physics

Electrostatics

Fields in Physics

Translational Dynamics

Study anywhere. Anytime. Across all devices.

Given $z = y^{2}$ - $2 x^{2}$ , Find ${(\frac{鈭� z}{鈭� x})}_{r}, {(\frac{鈭� z}{鈭� 胃})}_{x}, \frac{{鈭�}^{2} z}{鈭� x 鈭� 胃}$ .