- #1

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## Main Question or Discussion Point

Hello, I'm toying around with a Jacobian that has raised some interesting problems. It's a case of differentiating rates of some variable x, with respect to itself.

First one I suspect the answer is zero, though perhaps my reasoning is a bit flawed.

1.

[tex]

\frac{d}{d\theta}(\dot{\theta})

=\frac{d \dot{\theta}}{dt} \times \frac{dt}{d\theta}

=\ddot{\theta} \times \dot{\theta}^{-1}

=\ddot{\theta} / \dot{\theta}

=\frac{\Delta p }{\Delta t}}/\Delta p

=\Delta t \approx 0

[/tex]

The second I think you apply the total derivative rule to, but maybe not, should the angle and angle-rate be considered as two separate variables?

2.

[tex]

\frac{d}{d\theta}(\dot{\theta}cos\theta)

=\frac{dF}{d\theta}

=\frac{\partial F}{\partial \dot{\theta}} \times \frac{d\theta}{dt} +

\frac{\partial F}{\partial \theta} \times \ddot{\theta}

=\dot{\theta}cos\theta - \ddot{\theta}\dot{\theta}sin\theta

[/tex]

Last one has me flummaxed...

3.

[tex]

\frac{d}{d\theta}(\theta+\dot{\theta}dt)=?

[/tex]

And finally

4.

[tex]

\frac{d}{d\dot{\theta}}(q sin\phi tan\theta + r cos\phi tan\theta)

=\frac{1}{\ddot{\theta}}\times \frac{d}{dt}(q(t) sin\phi (t) tan\theta (t)+ r (t) cos\phi (t) tan\theta (t))

=?

[/tex]

Number 4 I arrive at from the chain rule (an example below):

[tex]

\frac{dy}{d\dot{\theta}}=\frac{dy}{dt} \times \frac{dt}{d\dot{\theta}}

=\frac{dy}{dt} \times \left(\frac{d\dot{\theta}}{dt}\right)^{-1}

=\frac{\dot{y}}{\ddot{\theta}}

[/tex]

Could anyone confirm what I've done so far (or point out any mistakes)? Cheers.

First one I suspect the answer is zero, though perhaps my reasoning is a bit flawed.

1.

[tex]

\frac{d}{d\theta}(\dot{\theta})

=\frac{d \dot{\theta}}{dt} \times \frac{dt}{d\theta}

=\ddot{\theta} \times \dot{\theta}^{-1}

=\ddot{\theta} / \dot{\theta}

=\frac{\Delta p }{\Delta t}}/\Delta p

=\Delta t \approx 0

[/tex]

The second I think you apply the total derivative rule to, but maybe not, should the angle and angle-rate be considered as two separate variables?

2.

[tex]

\frac{d}{d\theta}(\dot{\theta}cos\theta)

=\frac{dF}{d\theta}

=\frac{\partial F}{\partial \dot{\theta}} \times \frac{d\theta}{dt} +

\frac{\partial F}{\partial \theta} \times \ddot{\theta}

=\dot{\theta}cos\theta - \ddot{\theta}\dot{\theta}sin\theta

[/tex]

Last one has me flummaxed...

3.

[tex]

\frac{d}{d\theta}(\theta+\dot{\theta}dt)=?

[/tex]

And finally

4.

[tex]

\frac{d}{d\dot{\theta}}(q sin\phi tan\theta + r cos\phi tan\theta)

=\frac{1}{\ddot{\theta}}\times \frac{d}{dt}(q(t) sin\phi (t) tan\theta (t)+ r (t) cos\phi (t) tan\theta (t))

=?

[/tex]

Number 4 I arrive at from the chain rule (an example below):

[tex]

\frac{dy}{d\dot{\theta}}=\frac{dy}{dt} \times \frac{dt}{d\dot{\theta}}

=\frac{dy}{dt} \times \left(\frac{d\dot{\theta}}{dt}\right)^{-1}

=\frac{\dot{y}}{\ddot{\theta}}

[/tex]

Could anyone confirm what I've done so far (or point out any mistakes)? Cheers.