Q11-4P Verify the chain rule formulas聽... [FREE SOLUTION]

Chapter 1: Q11-4P (page 1)

Verify the chain rule formulas
$\frac{鈭� F}{鈭� x} = \frac{鈭� F}{鈭� r} \frac{鈭� r}{鈭� x} + \frac{鈭� F}{鈭� 胃} \frac{鈭� 胃}{鈭� x}$ ,
and similar formulas for
$\frac{鈭� F}{鈭� y}, \frac{鈭� F}{鈭� r}, \frac{鈭� F}{鈭� 胃}$ ,
using differentials. For example, write
$d F = \frac{鈭� F}{鈭� r} d r + \frac{鈭� F}{鈭� 胃} d 胃$
and substitute for $d r$ and $d 胃$ :
$d r = \frac{鈭� r}{鈭� x} d x + \frac{鈭� r}{鈭� y} d y$ (and similarly $d 胃$ ).
Collect coefficients of dx and dy; these are the values of $\frac{鈭� F}{鈭� x}$ and $\frac{鈭� F}{鈭� y}$ .

Short Answer

Expert verified

Hence, the required answers are:

$\frac{鈭� F}{鈭� x} = \frac{鈭� F}{鈭� r} \frac{鈭� r}{鈭� x} + \frac{鈭� F}{鈭� 胃} \frac{鈭� 胃}{鈭� x} \frac{鈭� F}{鈭� y} = \frac{鈭� F}{鈭� r} \frac{鈭� r}{鈭� y} + \frac{鈭� F}{鈭� 胃} \frac{鈭� 胃}{鈭� y} \frac{鈭� F}{鈭� r} = \frac{鈭� F}{鈭� x} \frac{鈭� x}{鈭� r} + \frac{鈭� F}{鈭� y} \frac{鈭� y}{鈭� r} \frac{鈭� F}{鈭� 胃} = \frac{鈭� F}{鈭� x} \frac{鈭� x}{鈭� 胃} + \frac{鈭� F}{鈭� y} \frac{鈭� y}{鈭� 胃}$

Step by step solution

Chain Rule

If any function $v = v (x, y)$ has independent variables which are also functions of some independent variables such that $x = x (s, t)$ and $y = y (s, t)$ , then the chain rule for the function is given by:

$\frac{鈭� v}{鈭� s} = \frac{鈭� v}{鈭� x} \frac{鈭� x}{鈭� s} + \frac{鈭� v}{鈭� y} \frac{鈭� y}{鈭� s} \frac{鈭� v}{鈭� t} = \frac{鈭� v}{鈭� x} \frac{鈭� x}{鈭� t} + \frac{鈭� v}{鈭� y} \frac{鈭� y}{鈭� t}$

Verify the differentials

The given formula is:

$\frac{鈭� F}{鈭� x} = \frac{鈭� F}{鈭� r} \frac{鈭� r}{鈭� x} + \frac{鈭� F}{鈭� 胃} \frac{鈭� 胃}{鈭� x}$

And we have:

$d F = \frac{鈭� F}{鈭� r} d r + \frac{鈭� F}{鈭� 胃} d 胃 . . . (1) d r = \frac{鈭� r}{鈭� x} d x + \frac{鈭� r}{鈭� y} d y . . . (2) d 胃 = \frac{鈭� 胃}{鈭� x} d x + \frac{鈭� 胃}{鈭� y} d y . . . (3)$

From equations (1), (2), and (3), we get:

$d F = \frac{鈭� F}{鈭� r} (\frac{鈭� r}{鈭� x} d x + \frac{鈭� r}{鈭� y} d y) + \frac{鈭� F}{鈭� 胃} (\frac{鈭� 胃}{鈭� x} d x + \frac{鈭� 胃}{鈭� y} d y) = (\frac{鈭� F}{鈭� r} \frac{鈭� r}{鈭� x} + \frac{鈭� F}{鈭� r} \frac{鈭� r}{鈭� y}) d x + (\frac{鈭� F}{鈭� 胃} \frac{鈭� 胃}{鈭� x} + \frac{鈭� F}{鈭� 胃} \frac{鈭� 胃}{鈭� y}) d y \frac{鈭� F}{鈭� x} = \frac{鈭� F}{鈭� r} \frac{鈭� r}{鈭� x} + \frac{鈭� F}{鈭� r} \frac{鈭� r}{鈭� y} \frac{鈭� F}{鈭� y} = \frac{鈭� F}{鈭� 胃} \frac{鈭� 胃}{鈭� x} + \frac{鈭� F}{鈭� 胃} \frac{鈭� 胃}{鈭� y}$

Verify the differentials

Similarly, using:

$d F = \frac{鈭� F}{鈭� x} d x + \frac{鈭� F}{鈭� y} d y . . . (3) d x = \frac{鈭� x}{鈭� r} d r + \frac{鈭� x}{鈭� 胃} d 胃 . . . (4) d y = \frac{鈭� y}{鈭� r} d r + \frac{鈭� y}{鈭� 胃} d 胃 . . . (5)$

We get:

$d F = \frac{鈭� F}{鈭� x} (\frac{鈭� x}{鈭� r} d r + \frac{鈭� x}{鈭� 胃} d 胃) + \frac{鈭� F}{鈭� y} (\frac{鈭� y}{鈭� r} d r + \frac{鈭� y}{鈭� 胃} d 胃) = (\frac{鈭� F}{鈭� x} \frac{鈭� x}{鈭� r} + \frac{鈭� F}{鈭� x} \frac{鈭� x}{鈭� 胃}) d r + (\frac{鈭� F}{鈭� y} \frac{鈭� y}{鈭� r} + \frac{鈭� F}{鈭� y} \frac{鈭� y}{鈭� 胃}) d 胃$

Therefore,

$\frac{鈭� F}{鈭� r} = \frac{鈭� F}{鈭� x} \frac{鈭� x}{鈭� r} + \frac{鈭� F}{鈭� y} \frac{鈭� y}{鈭� r} \frac{鈭� F}{鈭� 胃} = \frac{鈭� F}{鈭� x} \frac{鈭� x}{鈭� 胃} + \frac{鈭� F}{鈭� y} \frac{鈭� y}{鈭� 胃}$

Hence, the required answers are:

localid="1665130829170" $\frac{鈭� F}{鈭� x} = \frac{鈭� F}{鈭� r} \frac{鈭� r}{鈭� x} + \frac{鈭� F}{鈭� 胃} \frac{鈭� 胃}{鈭� x} \frac{鈭� F}{鈭� y} = \frac{鈭� F}{鈭� r} \frac{鈭� r}{鈭� y} + \frac{鈭� F}{鈭� 胃} \frac{鈭� 胃}{鈭� y} \frac{鈭� F}{鈭� r} = \frac{鈭� F}{鈭� x} \frac{鈭� x}{鈭� r} + \frac{鈭� F}{鈭� y} \frac{鈭� y}{鈭� r} \frac{鈭� F}{鈭� 胃} = \frac{鈭� F}{鈭� x} \frac{鈭� x}{鈭� 胃} + \frac{鈭� F}{鈭� y} \frac{鈭� y}{鈭� 胃}$

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Step by step solution

Chain Rule

Verify the differentials

Verify the differentials

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