Chapter 4: Q1P (page 196)

Consider a function $f (x, y)$ which can be expanded in a two-variable power series, (2.3) or (2.7). Let $x - a = h = ∆ x, y - b = k = ∆ y$ ; then localid="1664363195022" $x = a + ∆ x, y = b + ∆ y$ , so that $f (x, y)$ becomes $f (a + ∆ x, b + ∆ y)$ . The change $∆ z$ in $z = f (x + y)$ when x changes from a to $a + ∆ x$ and y changes from b to $b + ∆ y$ is then
$∆ z = f (a + ∆ x, b + ∆ y) - f (a, b)$ .
Use the series (2.7) to obtain (3.11) and to see explicitly what $\in_{1} a n d \in_{2}$ are and that they approach zero as $∆ x a n d ∆ y \to 0 a n d ∆ y \to 0$ .

Short Answer

Expert verified

Hence, we conclude $\in_{1}, \in_{2}$ tend to zero as both $∆ x a n d ∆ y$ tends to zero respectively.

Step by step solution

Form two variables power series

We have a function with $x = a + ∆ x a n d y = b + ∆ y$ subjected to change as:

$f (x, y) = f (a + ∆ x, b + ∆ y) = z$ $f (x, y) = f (a + ∆ x, b + ∆ y) = z$

And

$∆ z = f (a + ∆ x, b + ∆ y) - f (a, b) (1)$

Now, the two variable power series is given by:

$f (x, y) = \sum_{n = 0}^{\infty} {(h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y})}^{n} f (a, b) (2)$

Differentiate the given function

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

$∆ z = \sum_{n = 1}^{\infty} \frac{1}{n!} {(∆ x \frac{\partial}{\partial x} + ∆ y \frac{\partial}{\partial y})}^{n} f (a, b) - f (a, b) = ∆ x \frac{\partial f}{\partial x} + ∆ y \frac{\partial f}{\partial y} + \sum_{n = 1}^{\infty} \frac{1}{n!} {(∆ x \frac{\partial}{\partial x} + ∆ y \frac{\partial}{\partial y})}^{n} f (a, b) = ∆ x \frac{\partial f}{\partial x} + ∆ x \in_{1} + ∆ y \frac{\partial f}{\partial y} + ∆ x \in_{2} \{(x, y) \to 0 a s ∆ x, ∆ y \to 0\} = d z + ∆ x \in_{1} + ∆ y \in_{2}$

Define limit

From above evaluation, we obtained:

$∆ z = d z + ∆ x \in_{1} + ∆ y \in_{2}$

From this, we conclude:

$\in_{1}, \in_{2} \to 0, \forall ∆ x, ∆ y \to 0$

Hence, we conclude $\in_{1}, \in_{2}$ tend to zero as both $∆ x a n d ∆ y$ tends to zero respectively.

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Short Answer

Step by step solution

Form two variables power series

Differentiate the given function

Define limit

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