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answered Derivation of divergence in spherical coordinates from the divergence theorem
May
21
comment Scalar functions and manifolds
Sorry, I've reworded the edit. Hopefully that makes the answer clear.
May
21
comment Scalar functions and manifolds
Let me know if that edit is to your liking.
May
21
answered Scalar functions and manifolds
May
21
comment Scalar functions and manifolds
Are you sure he said a scalar function maps $M \mapsto \mathbb R^n$? Because one might expect that to be $M \mapsto \mathbb R$ instead.
May
20
answered Prove $\det(I + B) = 2(1 + tr(B)).$
May
19
revised how to rotate scaled-vector (orientation) by scaled-vector (rotation)
added a paragrpah on axis-angle approach
May
19
answered how to rotate scaled-vector (orientation) by scaled-vector (rotation)
May
19
comment how to rotate scaled-vector (orientation) by scaled-vector (rotation)
Hm, could you describe (or edit into your post) the numerical scheme you use to update orientations right now? What differential equations do the angular velocity and orientation obey, and how do you choose to evolve them?
May
19
comment how to rotate scaled-vector (orientation) by scaled-vector (rotation)
That being because a rotation of, say, a quaternion representing a rotation of 45 degrees is mathematically indistinguishable from one representing a rotation of 765 degrees.
May
19
comment how to rotate scaled-vector (orientation) by scaled-vector (rotation)
I see, so the issue isn't the rotation produced over the interval, it's that trying to convert back to recover your time derivatives is getting you fudged. Is that a fair assessment?
May
19
comment how to rotate scaled-vector (orientation) by scaled-vector (rotation)
Sorry, I'm not familiar with any other method that doesn't basically reduce to converting a quaternion back to axis-angle. To me, the "obvious" approach would be to use quaternions to keep the attitude of each object and to write the physics equations in terms of time derivatives of those quaternions. What issues did you have with that approach (I recognize you described this briefly, but I'm curious for a more detailed explanation)? What issues do you have with converting quats back and forth to axis-angle? I'm curious because such code may be slow due to trig, but shouldn't be messy.
May
11
comment Finding the basis one forms (covectors) corresponding to a particular formulation of basis vectors
When you write $\vec e_0 = x +y$, what do you mean by $x,y$? Are they vectors? Then how can you take partial derivatives with respect to them? Are they coordinates? Then how can a vector be equal to a sum of coordinates?
May
11
answered Curvature tensors and bivectors
May
8
answered Conceptual approach to the formula $\sum_{i=1}^n \det(v_1, \ldots, Av_i, \ldots, v_n)=\text{tr}(A)\det(v_1, \ldots, v_n)$
May
6
comment What is the mapping from purely imaginary quaternions to a vector in $\mathbb{R}^3$
There is no need to do such a thing. @Hayden has defined the function completely. To make an analogy, a function can be a black box: as long as you have a complete description of the output and input, there is no need to know what's going on "inside".
May
6
revised Can you multiply two vectors $v$ and $u$ using multiplication instead of dot or cross products
corrected some notation on tensor product
May
6
answered Can you multiply two vectors $v$ and $u$ using multiplication instead of dot or cross products
May
3
comment Dual tensor for partial derivative, if it has any meaning
A volume form on what manifold? $\mathbb R^3$? Something else?
May
2
answered Analogs to vectors — *unoriented* line segments