Q. 42 Find the gradient of the given f... [FREE SOLUTION]

Chapter 12: Q. 42 (page 961)

Find the gradient of the given functions in Exercises 37鈥�42.
$f (x, y, z) = \cos^{- 1} (\frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}})$

Short Answer

Expert verified

The gradient of the given function is $(\frac{z x}{x^{2} + y^{2} + z^{2} \sqrt{x^{2} + y^{2}}}, \frac{z y}{x^{2} + y^{2} + z^{2} \sqrt{x^{2} + y^{2}}}, - \frac{\sqrt{x^{2} + y^{2}}}{x^{2} + y^{2} + z^{2}})$ .

Step by step solution

Step 1. Given Information.

The given function is:

$f (x, y, z) = \cos^{- 1} (\frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}})$

Step 2. Calculation.

The gradient of the given function is:

$z = f (x, y, z) = \cos^{- 1} (\frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}}) 鈭� f (x, y, z) = [f_{x} (x, y, z), f_{y} (x, y, z), f_{z} (x, y, z)] - - - - - - - (1)$

Now find

$f_{x} (x, y, z) = \frac{鈭� f}{鈭� x} = \frac{- 1 [z {(- \frac{1}{2} (x^{2} + y^{2} + z^{2}))}^{\frac{- 3}{2}} 路 2 x]}{\sqrt{1 - \frac{z^{2}}{x^{2} + y^{2} + z^{2}}}} = \frac{z x}{x^{2} + y^{2} + z^{2} \sqrt{x^{2} + y^{2}}} f_{y} (x, y, z) = \frac{鈭� f}{鈭� y} = \frac{z y}{x^{2} + y^{2} + z^{2} \sqrt{x^{2} + y^{2}}} f_{z} (x, y, z) = \frac{鈭� f}{鈭� z} = \frac{- 1 [z {(- \frac{1}{2} (x^{2} + y^{2} + z^{2}))}^{\frac{- 3}{2}} 路 2 z + {(x^{2} + y^{2} + z^{2})}^{\frac{- 1}{2}} 路 1]}{\sqrt{1 - \frac{z^{2}}{x^{2} + y^{2} + z^{2}}}} = - \frac{\sqrt{x^{2} + y^{2}}}{x^{2} + y^{2} + z^{2}}$

Use these above values in (1) we get

$鈭� f (x, y, z) = (\frac{z x}{x^{2} + y^{2} + z^{2} \sqrt{x^{2} + y^{2}}}, \frac{z y}{x^{2} + y^{2} + z^{2} \sqrt{x^{2} + y^{2}}}, - \frac{\sqrt{x^{2} + y^{2}}}{x^{2} + y^{2} + z^{2}})$

Step 3. Conclusion.

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91影视

Find the gradient of the given functions in Exercises 37鈥�42.
$f (x, y, z) = \cos^{- 1} (\frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}})$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Calculation.

Step 3. Conclusion.

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91影视

Find the gradient of the given functions in Exercises 37鈥�42.fx,y,z=cos-11x2+y2+z2

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Calculation.

Step 3. Conclusion.

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Most popular questions from this chapter

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Find the gradient of the given functions in Exercises 37鈥�42.
$f (x, y, z) = \cos^{- 1} (\frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}})$