Q6MP If z=r2-x2, find聽(鈭倆鈭俽)胃,(... [FREE SOLUTION]

Chapter 4: Q6MP (page 238)

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

Short Answer

Expert verified

The value of ${(\frac{鈭� z}{鈭� r})}_{胃} = 2 r \sin^{2} 胃, {(\frac{鈭� z}{鈭� 胃})}_{r} = r^{2} \sin 2 胃, \frac{{鈭�}^{2} z}{鈭� r . 鈭� 胃} = 2 r \sin 2 胃 and {(\frac{鈭� z}{鈭� x})}_{y} = 0$

{(\frac{鈭� z}{鈭� r})}_{胃} = 2 r \sin^{2} 胃, {(\frac{鈭� z}{鈭� 胃})}_{r} = r^{2} \sin 2 胃, \frac{{鈭�}^{2} z}{鈭� r . 鈭� 胃} = 2 r \sin 2 胃 and {(\frac{鈭� z}{鈭� x})}_{y} = 0

{(\frac{鈭� z}{鈭� r})}_{胃} = 2 r \sin^{2} 胃, {(\frac{鈭� z}{鈭� 胃})}_{r} = r^{2} \sin 2 胃, \frac{{鈭�}^{2} z}{鈭� r . 鈭� 胃} = 2 r \sin 2 胃 and {(\frac{鈭� z}{鈭� x})}_{y} = 0

Step by step solution

Given Information

Given that, $z = r^{2} - x^{2}$ .

Formula Used

The value of $r^{2} = x^{2} + y^{2}$ ...(1)

x = r c o s 胃 y = r s i n 胃

Solving the given equation

The given equation

$z = r^{2} - x^{2}$ ...(3)

From equation 1 and 2,

$z = x^{2} + y^{2} - x^{2} z = y^{2}$

Substitute the value of y in the above equation.

$z = {(r \sin 胃)}^{2}$

Differentiate with respect to r,

$\frac{鈭� z}{鈭� r} = 2 r \sin^{2} 胃 {(\frac{鈭� z}{鈭� r})}_{胃} = 2 r \sin^{2} 胃$

We know that $z = r^{2} \sin^{2} 胃$

Differentiate with respect to $胃$

$\frac{鈭� z}{鈭� 胃} = 2 r^{2} \sin 胃 . \cos 胃 \frac{鈭� z}{鈭� 胃} = r^{2} \sin 2 胃 {(\frac{鈭� z}{鈭� 胃})}_{r} = r^{2} \sin 2 胃$

Solving further

$\frac{{鈭�}^{2} z}{鈭� r . 鈭� 胃} = \frac{鈭�}{鈭� r} (\frac{鈭� z}{鈭� 胃}) \frac{{鈭�}^{2} z}{鈭� r . 鈭� 胃} = \frac{鈭�}{鈭� r} (r^{2} \sin 2 胃) \frac{{鈭�}^{2} z}{鈭� r . 鈭� 胃} = 2 r \sin 2 胃$

We know that $z = y^{2}$

$\frac{鈭� z}{鈭� x} = 0 {(\frac{鈭� z}{鈭� x})}_{y} = 0$

Hence ${(\frac{鈭� z}{鈭� r})}_{胃} = 2 r \sin^{2} 胃, {(\frac{鈭� z}{鈭� 胃})}_{r} = r^{2} \sin 2 胃, \frac{{鈭�}^{2} z}{鈭� r . 鈭� 胃} = 2 r \sin 2 胃 and {(\frac{鈭� z}{鈭� x})}_{y} = 0$ .

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

Short Answer

Step by step solution

Given Information

Formula Used

Solving the given equation

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If z=r2-x2, find(鈭�z鈭�r)胃,(鈭�z鈭�胃)r,鈭�2z鈭�r鈭�胃,(鈭�z鈭�x)y

Short Answer

Step by step solution

Given Information

Formula Used

Solving the given equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

Geometrical and Physical Optics

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Famous Physicists

Study anywhere. Anytime. Across all devices.

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