Q 72. Prove that鈭噁(x,y)g(x,y)=g(x,y)... [FREE SOLUTION]

Chapter 12: Q 72. (page 965)

Prove that
$鈭� (\frac{f (x, y)}{g (x, y)}) = \frac{g (x, y) 鈭� f (x, y) - f (x, y) 鈭� g (x, y)}{(g (x, y))^{2}}$
where $g (x, y) 鈮� 0$

Short Answer

Expert verified

Solve for $鈭� (\frac{f (x, y)}{g (x, y)}) = (i \frac{鈭�}{鈭� x} + j \frac{鈭�}{鈭� y}) (\frac{f (x, y)}{g (x, y)})$ to prove the above relation.

Step by step solution

Given Information

Considering $f (x, y), g (x, y)$

We need to show that $鈭� (\frac{f (x, y)}{g (x, y)}) = \frac{g (x, y) 鈭� f (x, y) - f (x, y) 鈭� g (x, y)}{(g (x, y))^{2}}$

Use i∂∂x+j∂∂y for ∇

Substituting we get

$鈭� (\frac{f (x, y)}{g (x, y)}) = (i \frac{鈭�}{鈭� x} + j \frac{鈭�}{鈭� y}) (\frac{f (x, y)}{g (x, y)})$

$= i \frac{鈭�}{鈭� x} (\frac{f (x, y)}{g (x, y)}) + j \frac{鈭�}{鈭� y} (\frac{f (x, y)}{g (x, y)})$

Simplification

鈭� (\frac{f (x, y)}{g (x, y)}) = i [\frac{g (x, y) \frac{鈭�}{鈭� x} (f (x, y)) - f (x, y) \frac{鈭�}{鈭� x} (g (x, y))}{[g (x, y)]^{2}}]

$+ j [\frac{g (x, y) \frac{鈭�}{鈭� y} (f (x, y)) - f (x, y) \frac{鈭�}{鈭� y} (g (x, y))}{[g (x, y)]^{2}}]$

Rearranging, we get

$鈭� (\frac{f (x, y)}{g (x, y)}) = \frac{g (x, y) i \frac{鈭�}{鈭� x} (f (x, y)) + g (x, y) j \frac{鈭�}{鈭� y} (f (x, y))}{[g (x, y)]^{2}}$

$+ \frac{f (x, y) i \frac{鈭�}{鈭� x} (g (x, y)) + f (x, y) j \frac{鈭�}{鈭� y} (g (x, y))}{[g (x, y)]^{2}}$

$鈭� (\frac{f (x, y)}{g (x, y)}) = \frac{g (x, y) (i \frac{鈭�}{鈭� x} + j \frac{鈭�}{鈭� y}) (f (x, y))}{[g (x, y)]^{2}} + \frac{f (x, y) (i \frac{鈭�}{鈭� x} + j \frac{鈭�}{鈭� y}) (g (x, y))}{[g (x, y)]^{2}}$

$鈭� (\frac{f (x, y)}{g (x, y)}) = \frac{g (x, y) 路鈭� (f (x, y)) + f (x, y) 路鈭� (g (x, y))}{[g (x, y)]^{2}}$

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

Hence proved.

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91影视

Prove that
$鈭� (\frac{f (x, y)}{g (x, y)}) = \frac{g (x, y) 鈭� f (x, y) - f (x, y) 鈭� g (x, y)}{(g (x, y))^{2}}$
where $g (x, y) 鈮� 0$

Short Answer

Step by step solution

Given Information

Use i∂∂x+j∂∂y for ∇

Simplification

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91影视

Prove that鈭�f(x,y)g(x,y)=g(x,y)鈭�f(x,y)-f(x,y)鈭�g(x,y)(g(x,y))2whereg(x,y)鈮�0

Short Answer

Step by step solution

Given Information

Use i∂∂x+j∂∂y for ∇

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Pure Maths

Applied Mathematics

Statistics

Theoretical and Mathematical Physics

Calculus

Study anywhere. Anytime. Across all devices.

Prove that
$鈭� (\frac{f (x, y)}{g (x, y)}) = \frac{g (x, y) 鈭� f (x, y) - f (x, y) 鈭� g (x, y)}{(g (x, y))^{2}}$
where $g (x, y) 鈮� 0$