Problems 1) Derive eq. (329) from eqs. (3.21),(3.24)

To derive equation (3.29) from equations (3.21) and (3.24), we can use the following steps:<br /><br />1. Start with equation (3.21):<br /> \[ \text{Equation (3.21):} \quad \frac{d}{dt} \left( \frac{dy}{dx} \right) = \frac{d^2 y}{dx^2} \]<br /><br />2. Take the derivative of both sides of equation (3.21) with respect to \(x\):<br /> \[ \frac{d}{dx} \left( \frac{d}{dt} \left( \frac{dy}{dx} \right) \right) = \frac{d}{dx} \left( \frac{d^2 y}{dx^2} \right) \]<br /><br />3. Simplify the left side using the chain rule:<br /> \[ \frac{d^2}{dt \, dx} \left( \frac{dy}{dx} \right) = \frac{d^2 y}{dx^2} \]<br /><br />4. Substitute \( \frac{dy}{dx} \) with \( v \) (as in equation (3.24)):<br /> \[ \frac{d^2}{dt \, dx} \left( v \right) = \frac{d^2 y}{dx^2} \]<br /><br />5. Use equation (3.24) to express \( \frac{d^2 y}{dx^2} \):<br /> \[ \text{Equation (3.24):} \quad \frac{d^2 y}{dx^2} = \frac{d}{dt} \left( \frac{dv}{dx} \right) \]<br /><br />6. Substitute this into the equation from step 4:<br /> \[ \frac{d^2}{dt \, dx} \left( v \right) = \frac{d}{dt} \left( \frac{dv}{dx} \right) \]<br /><br />7. Recognize that \( \frac{d}{dt} \left( \frac{dv}{dx} \right) \) is the second derivative of \( v \) with respect to \( t \):<br /> \[ \frac{d^2}{dt \, dx} \left( v \right) = \frac{d^2 v}{dt \, dx} \]<br /><br />8. Therefore, we have:<br /> \[ \frac{d^2 v}{dt \, dx} = \frac{d}{dt} \left( \frac{dv}{dx} \right) \]<br /><br />This is equation (3.29).