James S. Cook
Reputation
10,727
Top tag
Next privilege 15,000 Rep.
Protect questions
 Nov 15 comment Why is $(-1) \cdot j = j \cdot (-1)$ for quaternions? @TravisBemrose I merely indicate the source of the notation for the usual vector algebra in modern texts. I agree that it is a dangerous game to play precisely for the reason my original comment indicated and for the reasons you outline; associativity in quaternions is not transferred to the cross-product. That said, I don't think it's at all crazy to suppose the quaternions are coming from the cross product in a certain sense. Perhaps if we look for a product which includes the cross and dot products in an associative fashion this leads us to quaternions... Nov 10 comment Tangent Spaces & Definition of Differentiation the $(-\epsilon, \epsilon)$ is just the domain of the curve. The equivalence class of curves with matching tangents can be taken to define the "tangent vector", or you can use derivations, or contravariant vectors... Nov 8 comment Why do mathematicans care so much about the incompressible Navier-Stokes equations? do mathematicians really care about this? Nov 8 comment The partial derivative of the Cross Product of Two Vectors? I found a linear algebra, well matrix algebra, text which has this at a computational level, cambridge.org/us/academic/subjects/economics/…. In this they support this sort of differentiation in a notation which more or less removes the theoretical burden (once you get used to the notation "vec" etc...). Anyway, nice answer. Nov 8 comment The partial derivative of the Cross Product of Two Vectors? I'm curious, where is this known? Do you happen to have a source text? Nov 4 revised Find all generators for the cyclic group $\mathbb{Z}_9 \times \mathbb{Z}_{10}$. edited title Nov 4 revised Geometric proof for triple vector product Jacobi identity grammar Oct 31 comment Mathematics of chemistry with focus on particular symmetries this is the sort of thing I think will fit his needs. Also, the Amazon reviews to this were helpful as they contain many additional pointers to other such books... Thanks. Oct 31 accepted Mathematics of chemistry with focus on particular symmetries Oct 31 comment Calculating $d(xdx + z^2dy + xydz)$ @John Doe you can look at supermath.info/math332mission8f2013soln.pdf see Problem 78 for another example. I try to weave a fair amount of these basic differential form calculations into my Advanced Calculus course. Oct 31 answered Calculating $d(xdx + z^2dy + xydz)$ Oct 31 comment Mathematics of chemistry with focus on particular symmetries Thanks for the recommendation. I'll hold off accepting for now, I don't see much explicit connection with Chemistry in the preview pages... of course, it is a beautiful book. Oct 30 asked Mathematics of chemistry with focus on particular symmetries Oct 28 comment Proof of Cauchy Riemann Equations in Polar Coordinates @Shemafied I suppose that would also be reasonable. Interesting. Oct 22 comment How to calculate the integral on surface which cannot be expressed in functions easily? Once you have a parametrization of a surface, say $X(u,v) = (x(u,v),y(u,v),z(u,v))$ then we calculate partial velocities $\frac{\partial X}{\partial u} = X_u$ and $\frac{\partial X}{\partial v} = X_v$. Of course, these are tangent to the coordinate curves $X(u,v_o)$ and $X(u_o,v)$ through $p = X(u_o,v_o)$. We have $N=X_u \times X_v$ is perpendicular to both $X_u$ and $X_v$ hence it must point in the normal direction to the surface. The $\langle \sin \theta, \cos \theta, 0 \rangle$ can be calculate as I indicate in the answer $N = X_{\theta} \times X_z$. It's a short calculation. Oct 22 comment How to calculate the integral on surface which cannot be expressed in functions easily? I think the calculation's ok now, however, this only accounts for the flux of the vector field through the cut-cylinder. If they intended the closed surface we still need to add the flux of the base ($z=0$) and cap ($z=2-x-y$). I think the flux through the base is zero since the normal component of the vector field is zero for the base. I'm not sure about the cap without calculation. Is $4\pi$ not the expected answer? Oct 22 answered How to calculate the integral on surface which cannot be expressed in functions easily? Oct 20 comment What does symmetry imply about the solution in mathematics? (Example: Gauss' law) You also wish to assume a linear dielectric media? Oct 20 comment What does symmetry imply about the solution in mathematics? (Example: Gauss' law) Ok, so what exactly do you mean by $\mathbf D$? Oct 20 comment What does symmetry imply about the solution in mathematics? (Example: Gauss' law) for specific mathematics of symmetries and conservation laws, we can study Noether's Theorem.