Chapter 2: Q36E (page 65)

Verify that $F (x, y)$ as defined by (9) and (10) satisfies conditions (4).

Short Answer

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Step by step solution

Differentiate with respect to x

The function $F (x, y)$ is given as

$\begin{matrix} F (x, y) = \int_{x_{o}}^{x} M (t, y) dt + g (y) \\ = N (x, y) - \frac{\partial}{\partial y} \int_{x_{o}}^{x} M (t, y) dt \end{matrix}$

To check if the function satisfies the conditions in equation (4), differentiate with respect to.

$\begin{matrix} \frac{\partial}{\partial x} F (x, y) = \frac{\partial}{\partial x} (\int_{x_{o}}^{x} M (t, y) dt + g (y)) \\ = \frac{\partial}{\partial x} (\int_{x_{o}}^{x} M (t, y) dt) + \frac{\partial}{\partial x} g (y) \end{matrix}$

Apply the Leibniz theorem then

$\frac{\partial}{\partial x} F (x, y) = M (x, y) + \frac{\partial}{\partial x} g (y)$

Since g(y) is function of only so

$\frac{\partial}{\partial x} F (x, y) = M (x, y)$

This is the condition in equation (4).

Find the result by applying the same procedure with respect to y

$\begin{matrix} \frac{\partial}{\partial y} F (x, y) = \frac{\partial}{\partial y} (\int_{x_{o}}^{x} M (t, y) dt + g (y)) \\ = \frac{\partial}{\partial y} (\int_{x_{o}}^{x} M (t, y) dt) + \frac{\partial}{\partial y} g (y) \\ \frac{\partial}{\partial x} F (x, y) = \frac{\partial}{\partial y} (\int_{x_{o}}^{x} M (t, y) dt) + g' (y) \end{matrix}$

The result $g' (y) = N (x, y) - \frac{\partial}{\partial y} (\int_{x_{o}}^{x} M (t, y) dt)$

$g' (y) = N (x, y) - \frac{\partial}{\partial y} (\int_{x_{o}}^{x} M (t, y) dt)$

Therefore, the result is $\begin{array}{l} \frac{\partial}{\partial x} F (x, y) = \frac{\partial}{\partial y} (\int_{x_{o}}^{x} M (t, y) dt) + N (x, y) - \frac{\partial}{\partial y} (\int_{x_{o}}^{x} M (t, y) dt) \\ \frac{\partial}{\partial x} F (x, y) = N (x, y) \end{array}$

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Verify that $F (x, y)$ as defined by (9) and (10) satisfies conditions (4).

Short Answer

Step by step solution

Differentiate with respect to x

Find the result by applying the same procedure with respect to y

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Verify that F (x,y)as defined by (9) and (10) satisfies conditions (4).

Short Answer

Step by step solution

Differentiate with respect to x

Find the result by applying the same procedure with respect to y

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Probability and Statistics

Theoretical and Mathematical Physics

Decision Maths

Pure Maths

Study anywhere. Anytime. Across all devices.

Verify that $F (x, y)$ as defined by (9) and (10) satisfies conditions (4).