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Trying2Learn

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- TL;DR Summary
- The derivative of a vector in a rotating frame (or even inertial)

I apologize: despite my verbosity, this is, I hope, a simple question.)

Consider the following relationship between a rotating reference frame and an inertial reference frame (both Bold), through a rotation matrix:

(the superscript is to designate the rotating frame e(1) and the I is for the inertial)

Now suppose I wish the time derivative of the rotating frame.

(I do not know how to put the dot overhead, sorry about this)

(The dot shoud be above the first

Now, did you see how I did not take the time derivative of the base frame?

Yes, PHYSICALLY, I know it is not rotating, so I do not have to do that.

But suppose I am a stubborn person (I am) who has memorized the "product rule" in calculus -- the first times the derivative of the second + the second times the derivative of the first.

And, now I see the product of a Rotation matrix (functions -- sine, cosine) and this bold base frame:

If I apply the rule, strictly, I put the dot over it and assert the result is zero.

But suppose I want to be stubborn and say: "but that base frame is not a function! I only know how to use the product rule for functions"

How do I answer myself?

The ONLY guess I can make is the following:

We began with coordinate functions: x1(1), x2(t) and x3(t)

We take time derivatives as we move a point, P(x1,x2, x3)

∂P(x1,x2, x3)/∂x1,

∂P(x1,x2, x3)/∂x2,

∂P(x1,x2, x3)/∂x3

Then, we decide to create the frame by dropping the point P, notation and looking at only the partial notation

We do this like Ted Frankel did on page (3) of this:

http://www.math.ucsd.edu/~tfrankel/the_geometry_of_physics.pdf

And while we have a frame, its three axes "began their lives as derivatives of coordinate FUNCTIONS, so we CAN use the product rule."

Can anyone advise a better way?

For example, suppose BOTH frames are moving and we want the derivative. We have to use the product rule and get the following sequence.

(and please forgive me for the placement of the dot--all dots should be directly overhead)

Again, how do I know (in my ignorance and stupidity), to apply a product rule (proven for functions) to a base frame? (I have no issue, by the way, with the matrix structure formulation of the time derivative -- that is not an issue).

(And if you can tell me how to get the dot directly overhead, that would be nice, too.)

Consider the following relationship between a rotating reference frame and an inertial reference frame (both Bold), through a rotation matrix:

(the superscript is to designate the rotating frame e(1) and the I is for the inertial)

**e**^{(1)}=**e**^{I}R^{(1)}Now suppose I wish the time derivative of the rotating frame.

(I do not know how to put the dot overhead, sorry about this)

(The dot shoud be above the first

**e**and above the R)^{.}**e**^{(1)}=**e**^{I}^{.}R^{(1)}Now, did you see how I did not take the time derivative of the base frame?

Yes, PHYSICALLY, I know it is not rotating, so I do not have to do that.

But suppose I am a stubborn person (I am) who has memorized the "product rule" in calculus -- the first times the derivative of the second + the second times the derivative of the first.

And, now I see the product of a Rotation matrix (functions -- sine, cosine) and this bold base frame:

**e**^{I}If I apply the rule, strictly, I put the dot over it and assert the result is zero.

But suppose I want to be stubborn and say: "but that base frame is not a function! I only know how to use the product rule for functions"

How do I answer myself?

The ONLY guess I can make is the following:

We began with coordinate functions: x1(1), x2(t) and x3(t)

We take time derivatives as we move a point, P(x1,x2, x3)

∂P(x1,x2, x3)/∂x1,

∂P(x1,x2, x3)/∂x2,

∂P(x1,x2, x3)/∂x3

Then, we decide to create the frame by dropping the point P, notation and looking at only the partial notation

**e**^{(1)}≡∂/∂x1,**e**^{(2)}≡∂/∂x2,**e**^{(3)}≡∂/∂x3We do this like Ted Frankel did on page (3) of this:

http://www.math.ucsd.edu/~tfrankel/the_geometry_of_physics.pdf

And while we have a frame, its three axes "began their lives as derivatives of coordinate FUNCTIONS, so we CAN use the product rule."

Can anyone advise a better way?

For example, suppose BOTH frames are moving and we want the derivative. We have to use the product rule and get the following sequence.

**e**^{(2)}=**e**^{(1)}R^{(2/1)}(and please forgive me for the placement of the dot--all dots should be directly overhead)

^{.}e^{(2)}=^{.}e^{(1)}R^{(2/1)}+**e**^{(1)}**R**^{.}^{(2/1)}Again, how do I know (in my ignorance and stupidity), to apply a product rule (proven for functions) to a base frame? (I have no issue, by the way, with the matrix structure formulation of the time derivative -- that is not an issue).

(And if you can tell me how to get the dot directly overhead, that would be nice, too.)