# Evaluate partial derivative. chain rule?

1. Oct 10, 2012

### 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. Unfortunatly I just cant 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..

Last edited: Oct 10, 2012
2. Oct 11, 2012

### 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)

3. Oct 11, 2012

### james1234

Re: Evaluate partial derivative. chain rule??

Hi Tim,

well if it cant 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 refering 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 dependant on theta, unfortunatly as far as I can tell I dont 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!!

4. Oct 11, 2012

### 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

5. Oct 28, 2012

### james1234

Re: Evaluate partial derivative. chain rule??

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 lenght 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 independant 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 independant 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 independant varibles as listed in my initial problem)

Thank you!!! Apologies for the lenght of my post!

Last edited: Oct 28, 2012
6. Oct 28, 2012

### tiny-tim

hi james1234!
yes! V and θ are the usual polar coordinates (which we normally call r and θ) …

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 Veθ)