40. If f(x, y) is continuous over R: a < x < b, c < y < d andF(x, y) = | | flu, v) dv duon the interior of R, find the second partial derivatives Fy and Fr.

Given that if f(x, y) is continuous over R : a≤x≤b, c≤y≤d and F(x,…

Answered: 40. If f(x, y) is continuous over R: a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

40. If f(x, y) is continuous over R: a < x < b, c < y < d and
F(x, y) = | | flu, v) dv du
on the interior of R, find the second partial derivatives Fy and Fr.

Expert Solution

Step 1

Given that if $f (x, y)$ is continuous over $R : a \leq x \leq b, c \leq y \leq d$ and

$F (x, y) = \int_{a}^{x} \int_{c}^{y} f (u, v) d v d u,$

on the interior of $R .$

Step 2

Now,

$F (x, y) = \int_{a}^{x} \int_{c}^{y} f (u, v) d u d v .$

Assume that,

$\int_{c}^{y} f (u, v) d u = h (y, v) .$

This implies that,

$F (x, y) = \int_{a}^{x} h (y, v) d v,$

Differentiating both sides with respect to $' y',$ we obtain

$\frac{\partial}{\partial y} (F (x, y)) = \frac{\partial}{\partial y} (\int_{a}^{x} h (y, v) d v), = \int_{a}^{x} \frac{\partial}{\partial y} (h (y, v)) d v, = \int_{a}^{x} \frac{\partial}{\partial y} (\int_{c}^{y} f (x, y) d u) d v, \{∵ h (y, v) = \int_{c}^{y} f (x, y) d u\} = \int_{a}^{x} (\int_{c}^{y} \frac{\partial}{\partial y} f (x, y) d u + f (y, v)) d v \Rightarrow \frac{\partial}{\partial y} (F (x, y)) = \int_{a}^{x} f (y, v) d v .$