Chapter 2: Q35E (page 65)

Using condition (5), show that the right-hand side of (10) is independent of x by showing that its partial derivative with respect to x is zero. [Hint: Since the partial derivatives of M are continuous, Leibniz’s theorem allows you to interchange the operations of integration and differentiation.]

Short Answer

Expert verified

Leibniz theorem allows interchanging the operations of integration and Differentiation.

Step by step solution

Show that RHS of equation (10) is independent of x

The condition of exactness is $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ .

Now, take equation (10) and partially differentiate the RHS w.r.t. x

$\begin{array}{l} \frac{\partial}{\partial x} (RHS) = \frac{\partial}{\partial x} (N (x, y) - \frac{\partial}{\partial x} \frac{\partial}{\partial y} \int_{x_{o}}^{x} M (t, y) dt) \\ = \frac{\partial N (x, y)}{\partial x} - \frac{\partial}{\partial y} (\frac{\partial}{\partial x} \int_{x_{o}}^{x} M (t, y) dt) \\ = \frac{\partial N (x, y)}{\partial x} - \frac{\partial}{\partial y} (\int_{x_{o}}^{x} \frac{\partial}{\partial x} M (t, y) dt) \end{array}$

Apply Leibniz rule

Apply the Leibniz rule, then

$\frac{\partial}{\partial x} (RHS) = \frac{\partial N (x, y)}{\partial x} - \frac{\partial}{\partial y} M (x, y)$

Now, uses the condition (5) then

$\frac{\partial}{\partial x} (N (x, y) - \frac{\partial}{\partial x} \frac{\partial}{\partial y} \int_{x_{o}}^{x} M (t, y) dt) = 0$

Therefore the RHS of the equation (10) is independent of x.

Hence, that the right-hand side of (10) is independent of x

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Using condition (5), show that the right-hand side of (10) is independent of x by showing that its partial derivative with respect to x is zero. [Hint: Since the partial derivatives of M are continuous, Leibniz’s theorem allows you to interchange the operations of integration and differentiation.]

Short Answer

Step by step solution

Show that RHS of equation (10) is independent of x

Apply Leibniz rule

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