Dear DH. We in general cannot know a system by knowing its parts. Systems take on properties of their own. This is why we can build and program computers. Yes at one level they are arrays of transitors, but understanding the transitor arrays does not help us understand the accounting system...
The answer to this question is precisely the same reason we can't predict the weather. Differentiation essentially makes things into linear functions. Just because we understand all there is to know about the laws of motion in a tiny piece of fluid, does not imply we can understand the entire...
I think the small angle commutativity of rotation allows that change between the vector world of angular velocity, and the non vector world of rotation. By definition differentiation is about deltas approaching zero. Which makes the differential linear and commutative where the parent function...
I think I understand. A vector must follow the laws of vector addition. If we had 3 axes, representing the three possible orthogonal axes of rotation. each axis marked out with angle of rotation along its length, each point in that three dimensional space would be a specific rotation from the...
You said
Integrating angular velocity with respect to time obviously does result in a vector. The problem is that this integrated vector in general does not have any physical meaning.
The comment about physical meaning was yours, my comment was agreeing with you. I have shown quite...
In general a rotational vector may not have physical meaning, but in rotational kinematics it is the rotational analogue of distance, and can be used in equations accordingly. WorkDone = Torque.Rotation. I accept that this is a special case, and that the more normal understanding of rotation...
I accept that in general rotation is non-commutative. My question involves a number of different aspects. (1) The calculus aspect: if the integration of an angular acceleration to an angular velocity results in a colinear vector, why would the same operation, that is integration with respect...
Is that the essential argument, rotation is in the wrong type of space to be a vector, that vectors exist only in eucledian spaces? Rotation is obviously not a scalar because of its axial direction, if it is not a vector, scalar or field, then what is it?
The earlier post about commutivity and turning the book was defined by operations in eucledian space. I suggest that the reason that the operations were not commutative was the choice of a rectangular coordinate system not appropriate for the discussion. Instead we should consider a spherical...
I have gone to Hamiltons "Elements of Quarternions" and angles the way I have defined them are vectors according to his definition, having direction and magnitude.
Okay I have done some experiments and rotation does not follow cross product rules either. For a cross product (a X b)=-(b X a) and it does not work.
I will read up pseudovectors on wikipedia as you suggest.
If we agree that angular velocity and angular acceleration are vectors, the it can be seen that the differentiation by time results in a colinear vector with the one differentiated by. It would follow that integration would also result in a colinear vector as is true from angular acceleration...
I do not expect the application of a series of rotations of an object in 3d space to be commutative because they are cross product operations and cross products are not commutative. Therefore I do not accept your argument as it seems to be based on commutativity.
As a programmer I understand this desire to keep things to a single formula. In code it is good to test for zero on division and make a judgement on the interpretation of the situation. In terms of this problem, if the calculations were using IEEE 754, there are representations for positive...
Hello
I have been thinking about dimensional analysis with respect to computer systems. It has become obvious to me that to be meaningful such an analysis has to distinguish dot product from vector products. An area can be considered the cross product of vectors. The question that arose is...