Q6P By expandingw(x,y,z)聽聽in a th... [FREE SOLUTION]

Chapter 15: Q6P (page 775)

By expanding $w (x, y, z)$ in a three-variable power series similarto , $(10.10)$ show that
$r_{w} = \sqrt{{(\frac{鈭� w}{鈭� x})}^{2} r_{x}^{2} + {(\frac{鈭� w}{鈭� y})}^{2} r_{y}^{2} + {(\frac{鈭� w}{鈭� z})}^{2} r_{z}^{2}}$

Short Answer

Expert verified

Required expressions are,

\begin{matrix} E [w (x, y, z)] = w (渭_{x}, 渭_{y}, 渭_{s}) \\ Var (\overset{炉}{w}) = {(\frac{鈭� w}{鈭� x})}^{2} 蟽_{m x}^{2} + {(\frac{鈭� w}{鈭� y})}^{2} 蟽_{m y}^{2} + {(\frac{鈭� w}{鈭� z})}^{2} 蟽_{m x}^{2} \\ 蟽_{m 鈭�} = \sqrt{{(\frac{鈭� w}{鈭� x})}^{2} 蟽_{m x}^{2} + {(\frac{鈭� w}{鈭� y})}^{2} 蟽_{m y}^{2} + {(\frac{鈭� w}{鈭� z})}^{2} 蟽_{m z}^{2}} \\ r_{w} = \sqrt{{(\frac{鈭� w}{鈭� x})}^{2} r_{x}^{2} + {(\frac{鈭� w}{鈭� y})}^{2} r_{y}^{2} + {(\frac{鈭� w}{鈭� z})}^{2} r_{z}^{2}} \end{matrix}

Step by step solution

Given Information

The equation, $(10.10)$ i.e.,role="math" localid="1664361601057" $w (x, y) 鈮� w (渭_{x}, 渭_{y}) + (\frac{鈭� w}{鈭� x}) (x 鈭� 渭_{x}) + (\frac{鈭� w}{鈭� y}) (y 鈭� 渭_{y})$

Definition of Power series.

An infinite series is a polynomial with an infinite number of elements that can be thought of as a power series.

Step 3:Expand function w with Taylor’s series.

$w (x, y, z) 鈮� w (渭_{x}, 渭_{y}) + (\frac{鈭� w}{鈭� x}) (x 鈭� 渭_{x}) + (\frac{鈭� w}{鈭� y}) (y 鈭� 渭_{y}) + (\frac{鈭� w}{鈭� z}) (z 鈭� 渭_{z})$

Calculate mean and variance value of function w.

Calculate expected value.

$\begin{matrix} E [w (x, y, z)] 鈮� w (渭_{x}, 渭_{y}, 渭_{z}) + (\frac{鈭� w}{鈭� x}) [E (x) 鈭� 渭_{x}] + (\frac{鈭� w}{鈭� y}) [E (y) 鈭� 渭_{y}] + (\frac{鈭� w}{鈭� z}) [E (z) 鈭� 渭_{z}] \\ = w (渭_{x}, 渭_{y}, 渭_{s}) \end{matrix}$

Calculate variance.

$\begin{matrix} Var (\overset{炉}{w}) = Var [w (\overset{炉}{x}, \overset{炉}{y}, \overset{炉}{z})] \\ = Var [w (渭_{x}, 渭_{y}, 渭_{z}) + (\frac{鈭� w}{鈭� x}) (\overset{炉}{x} 鈭� 渭_{z}) + (\frac{鈭� w}{鈭� y}) (\overset{炉}{y} 鈭� 渭_{y}) + (\frac{鈭� w}{鈭� z}) (\overset{炉}{z} 鈭� 渭_{z})] \\ = {(\frac{鈭� w}{鈭� x})}^{2} 蟽_{m x}^{2} + {(\frac{鈭� w}{鈭� y})}^{2} 蟽_{m y}^{2} + {(\frac{鈭� w}{鈭� z})}^{2} 蟽_{m x}^{2} \end{matrix}$

Calculate standard error.

Calculate standard error value.

$\begin{matrix} 蟽_{m 鈭�} = \sqrt{Var (\overset{炉}{w})} \\ = \sqrt{{(\frac{鈭� w}{鈭� x})}^{2} 蟽_{m x}^{2} + {(\frac{鈭� w}{鈭� y})}^{2} 蟽_{m y}^{2} + {(\frac{鈭� w}{鈭� z})}^{2} 蟽_{m z}^{2}} \end{matrix}$

Calculate probable error with confidence interval.

Calculate probable error value.

$\begin{matrix} r_{w} = I 蟽_{m w} \\ = I \sqrt{{(\frac{鈭� w}{鈭� x})}^{2} 蟽_{m x}^{2} + {(\frac{鈭� w}{鈭� y})}^{2} 蟽_{m y}^{2} + {(\frac{鈭� w}{鈭� z})}^{2} 蟽_{m z}^{2}} \\ = \sqrt{{(\frac{鈭� w}{鈭� x})}^{2} {(I 蟽_{m x})}^{2} + {(\frac{鈭� w}{鈭� y})}^{2} {(I 蟽_{m y})}^{2} + {(\frac{鈭� w}{鈭� z})}^{2} {(I 蟽_{m z})}^{2}} \\ = \sqrt{{(\frac{鈭� w}{鈭� x})}^{2} r_{x}^{2} + {(\frac{鈭� w}{鈭� y})}^{2} r_{y}^{2} + {(\frac{鈭� w}{鈭� z})}^{2} r_{z}^{2}} \end{matrix}$

Hence, required expressions are,

$\begin{matrix} E [w (x, y, z)] = w (渭_{x}, 渭_{y}, 渭_{s}) \\ Var (\overset{炉}{w}) = {(\frac{鈭� w}{鈭� x})}^{2} 蟽_{m x}^{2} + {(\frac{鈭� w}{鈭� y})}^{2} 蟽_{m y}^{2} + {(\frac{鈭� w}{鈭� z})}^{2} 蟽_{m x}^{2} \\ 蟽_{m 鈭�} = \sqrt{{(\frac{鈭� w}{鈭� x})}^{2} 蟽_{m x}^{2} + {(\frac{鈭� w}{鈭� y})}^{2} 蟽_{m y}^{2} + {(\frac{鈭� w}{鈭� z})}^{2} 蟽_{m z}^{2}} \\ r_{w} = \sqrt{{(\frac{鈭� w}{鈭� x})}^{2} r_{x}^{2} + {(\frac{鈭� w}{鈭� y})}^{2} r_{y}^{2} + {(\frac{鈭� w}{鈭� z})}^{2} r_{z}^{2}} \end{matrix}$

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

By expanding $w (x, y, z)$ in a three-variable power series similarto , $(10.10)$ show that
$r_{w} = \sqrt{{(\frac{鈭� w}{鈭� x})}^{2} r_{x}^{2} + {(\frac{鈭� w}{鈭� y})}^{2} r_{y}^{2} + {(\frac{鈭� w}{鈭� z})}^{2} r_{z}^{2}}$

Short Answer

Step by step solution

Given Information

Definition of Power series.

Step 3:Expand function w with Taylor’s series.

Calculate mean and variance value of function w.

Calculate standard error.

Calculate probable error with confidence interval.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Quantum Physics

Physics of Motion

Medical Physics

Engineering Physics

Famous Physicists

Fundamentals of Physics

Study anywhere. Anytime. Across all devices.

91影视

By expandingw(x,y,z) in a three-variable power series similarto ,(10.10)show thatrw=(鈭�w鈭�x)2rx2+(鈭�w鈭�y)2ry2+(鈭�w鈭�z)2rz2

Short Answer

Step by step solution

Given Information

Definition of Power series.

Step 3:Expand function w with Taylor’s series.

Calculate mean and variance value of function w.

Calculate standard error.

Calculate probable error with confidence interval.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Quantum Physics

Physics of Motion

Medical Physics

Engineering Physics

Famous Physicists

Fundamentals of Physics

Study anywhere. Anytime. Across all devices.

By expanding $w (x, y, z)$ in a three-variable power series similarto , $(10.10)$ show that
$r_{w} = \sqrt{{(\frac{鈭� w}{鈭� x})}^{2} r_{x}^{2} + {(\frac{鈭� w}{鈭� y})}^{2} r_{y}^{2} + {(\frac{鈭� w}{鈭� z})}^{2} r_{z}^{2}}$