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Chapter 12: Q 63. (page 965)

Prove that $鈥� 鈭� f (x, y, z) 鈥� = 1$ for every point in the domain of the function $f (x, y, z) = \sqrt{x^{2} + y^{2} + z^{2}}$ except the origin

Short Answer

Expert verified

Solving $鈭� f = i \frac{鈭� f}{鈭� x} + j \frac{鈭� f}{鈭� y} + k \frac{鈭� f}{鈭� z}$ will prove the result.

Step by step solution

01

Given Information

It is given that

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

02

Taking the divergence

Solving LHS

$鈥� 鈭� f (x, y, z) 鈥�$

$鈭� f = i \frac{鈭� f}{鈭� x} + j \frac{鈭� f}{鈭� y} + k \frac{鈭� f}{鈭� z}$

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

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

and

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

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

also

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

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

03

Simplification

Solving, we get

$鈥� 鈭� f (x, y, z) 鈥� = ||i \frac{鈭� f}{鈭� x} + j \frac{鈭� f}{鈭� y} + k \frac{鈭� f}{鈭� z}||$

$鈥� 鈭� f (x, y, z) 鈥� = ||i (\frac{x}{\sqrt{x^{2} + y^{2} + z^{2}}}) + j (\frac{y}{\sqrt{x^{2} + y^{2} + z^{2}}}) + k (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}})||$

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

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

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

$鈥� 鈭� f (x, y, z) 鈥� = 1$

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Most popular questions from this chapter

Given a function of n variables, $z = f (x_{1}, x_{2}, 鈥�, x_{n}),$ and a constraint equation, $g (x_{1}, x_{2}, 鈥�, x_{n}) = 0,$ how many equations would we obtain if we tried to optimize f by the method of Lagrange multipliers?

Evaluate the following limits, or explain why the limit does not exist.

$\lim_{(x, y) 鈫� (0, 0)} \frac{x + y}{x^{2} + y^{2}}$

$A chain rule for functions of two variables : Let f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t . Make a conjecture about the relationship between \frac{鈭� f}{鈭� s}, \frac{鈭� f}{鈭� x}, \frac{鈭� f}{鈭� y}, \frac{鈭� x}{鈭� s}, and \frac{鈭� y}{鈭� s} .$

Use Theorem 12.32 to find the indicated derivatives in Exercises 21鈥�26. Express your answers as functions of a single variable.

$\frac{d 胃}{d t} w h e n 胃 = \tan^{鈭� 1} \frac{y}{x}, x = e^{t}, a n d y = e^{2 t}$

In Exercises 24鈥�32, find the maximum and minimum of the functionf subject to the given constraint. In each case explain why the maximum and minimum must both exist.

$f (x, y, z) = x y z when x^{2} + 4 y^{2} + 16 z^{2} = 64$

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