Evaluate partial derivative. chain rule?

[james1234]

Evaluate partial derivative. chain rule??

I would like to represent the term identified in the image as (term 1)

in terms of those partial derviatives that are known. unfortunately I just can't seem to wrap my head around it at the moment.

A prod in the right direction would be greatly appreciated.

It would appear that V^2 is also equal to u^2+w^2..

 

[tiny-tim]

hi james1234!

(try using the X² button just above the Reply box )

i don't see how you can answer that without knowing what the other variable is

∂/∂θ has no meaning unless you specify what the other variable is, because ∂/∂θ is defined as being evaluated with the other variable kept constant

(if the other variable was r, as in the usual polar coordinates, you could then work out ∂u/∂θ and ∂v/∂θ)

 

[james1234]

Hi Tim,

thanks for your reply!

well if it can't be solved then that does make me feel a little bit better.. A tiny bit anyway.

However, when you say "if the other variable was r" are your referring to the relation V² = u²+w².. (thanks for the tip)

Just in case a little more perspective would be helpful, I have defined V as the velocity of the air traveling around my model aeroplane, where u is the velocity in the x axis, w the velocity in the z axis and theta is the angle of attack (for a small perturbation).

Now, I know that V is dependent on theta, unfortunately as far as I can tell I don't have a simple equation relating the change in velocity due to a change in the angle of attack i.e. u(θ),w(θ) or better yet V(θ).

I do have a moment equation for the angular rate \dot{θ} in terms of both V and theta but I can't see how this would help. Or am I missing the point entirely..?

So having linearized the equations of motion in the longitudinal plane with respect to u,w,theta,\dot{θ} (taking the first term of the taylor series expansion) I end up with an expresion written in terms of

where i know that

So my problem. Can I somehow evaluate the expressions

to make it look all neat and tidy?? I hope so!

Thanks again for your help!

 

[tiny-tim]

hi james1234!

(have a curly d: ∂ )

do you mean u = |v|cosθ, w = |v|sinθ, u² + w² = |v|² ?

then the two pairs of coordinates are u and w, or |v| and θ

you can't have ∂|v|/∂θ, they're in the same pair

 

[james1234]

Hi Tim,

I've been pondering your response for the past week or two. Clearly I've failed to grasp the fundamentals.

Regarding your statement "you can't have ∂|v|/∂θ, they're in the same pair"??
Does this mean that you do not consider V to vary as a function of theta?! Now I'm really confused!

Ok back to basics! In terms of a really simple trig problem:

We know that the angle and magnitude of V must change as a function of W (as the magnitude of W changes ~ and theta..) with U held constant. Likewise we know that the angle and mangnitude of V will change as the length of U (and angle theta) undergoes change.
Does not V also change as a function of theta??!
Certainly as the angle of V changes (change in theta) either U or W must also change.. Is this what your suggesting? that as a change in theta is coupled to a change in either U or W that the partial derivative of V with respect to theta does not exist (more than one independent variable is undergoing change)?
Surely I can only ever 'fix' one of these variables (U,W,theta).

As far as I can see I have three independent variables (U V theta) for which any change in V signals a change in a either (U and W, U and theta or W and theta).
So assuming V is a vector representing the magnitude and angle of the air stream (free stream) traveling over the surface of my model aeroplane (and U and W are the respective components of this vector) can I identify the partial derivative of V with respect to theta?? I'm inclined to think so! (no??)

Could you shed a little more light on why one might not obtain a representation for ∂V/∂θ? and consequently a simple expression for ∂V/∂θ that will make my equations nice and neat (i.e. represent ∂V/∂θ in terms of one the independent varibles as listed in my initial problem)

Thank you! Apologies for the length of my post!

 

[tiny-tim]

hi james1234!

Does this mean that you do not consider V to vary as a function of theta?!

yes! V and θ are the usual polar coordinates (which we normally call r and θ) …

r is not a function of θ, they're independent variables, aren't they?
We know that the angle and magnitude of V must change as a function of W (as the magnitude of W changes ~ and theta..) with U held constant. Likewise we know that the angle and mangnitude of V will change as the length of U (and angle theta) undergoes change.
Does not V also change as a function of theta??!

you see to be confusing the vector V, which can be represented as (V,θ), with the scalar V

the question is about ∂V/∂θ, not ∂V/∂θ

(for what it's worth, ∂V/∂θ is the vector Ve_θ)