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comment Curl: invariant under change of basis or not?
Ok, let's impose a distinction between coordinates (which are the arguments for a scalar or vector field) and components (which are individual scalar functions that, along with a basis, describe a vector field). - With that in mind, you could have a transformation change only the components (e.g. a local rotation) without the coordinates, or you could have a transformation that changes the coordinates and the components (because any map that changes the coordinates also defines a new coordinate basis). So which do you want? change coordinates and evaluate wrt the new coord basis? Or not?
1d
comment Curl: invariant under change of basis or not?
Your wording is somewhat imprecise. You refer to a "change of basis" throughout most of this question, but then you say the coordinates would change. These concepts are different: do you mean a change of coordinates or not? Or do you mean merely (and only) a change of basis?
Jul
1
comment Advanced calculus: Solving quaternion differential equations
You basically have the most general system imaginable here. Is this the actual problem you have, or do you have something more specific?
Jun
30
comment Trouble expanding a del operator expression
Yes. You have incorrectly applying the product rule. In your last expression, both those terms are equal, and you get a difference of a factor of 2 compared to the correct expression.
Jun
30
comment Trouble expanding a del operator expression
Technically, yes, but to me the left-hand side offers no substantive difference in meaning or advantage for computation. Conceptually, that's taking the derivative of a differential operator--it makes no sense.
Jun
30
comment Trouble expanding a del operator expression
Partial derivatives are associative? Do you mean that they commute?
Jun
30
comment Trouble expanding a del operator expression
You seem confused. $\dot \nabla \cdot \dot \nabla$ is nonsense. $\ddot \nabla$ is also nonsense. Your results should be consistent with the notion that $\dot \nabla \cdot [(\dot u \cdot \nabla) u] = (\partial_i u^j) \partial_j u^i$.
Jun
28
comment Numerically find a potential field from gradient
How, or in what way, do you obtain the values of the vector field on the grid? Are they given to you, or is there some other quantity from which you can measure the vector field?
Jun
26
comment Divergence of inverse square vector field
I assume you mean why there's an $r^2$ in the derivative when you do the divergence using spherical coordinates, but no such term when you compute divergence in cartesian. Using the integral definition of divergence, you would see that the $r^2$ comes from the surface element at constant $r$: $r^2 \sin \theta \, d\theta \, d\phi$. The $\sin \theta$ drops when looking at the radial derivative, but the $r^2$ remains. The formulas for divergence and other such derivatives are usually provided in reference material for any vector calculus textbook, so you don't have to derive them yourself.
Jun
25
comment Divergence of inverse square vector field
You thought that the divergence in spherical coordinates is just a partial derivative with respect to $r$ for the radial part, and it's not. The correct form (with $r^2$ running around) is what I stated, and it can be derived using the integral form that I used throughout this answer, if you're curious how that comes to be.
Jun
25
answered Divergence of inverse square vector field
Jun
23
comment Hodge self-duality in Minkowski spacetime
Fair enough, just let me know where and when.
Jun
23
comment Hodge self-duality in Minkowski spacetime
I'm open to suggestion regarding how to go about that.
Jun
22
revised Hodge self-duality in Minkowski spacetime
added a section using hodge star notation
Jun
22
answered Hodge self-duality in Minkowski spacetime
Jun
22
comment Hodge self-duality in Minkowski spacetime
Nevermind. By "null form" you mean the zero 2-form?
Jun
21
comment Hodge self-duality in Minkowski spacetime
I think you mean $(\star \omega)_{23} = \omega_{10}$?
Jun
17
comment Is it permissible to factor out a dot product?
What ultimately would you want to prove if you could make some statement about $y$ relative to $x$?
Jun
5
answered How to visualize rotation on a hyperbola?
Jun
5
revised Trigonometric Differentiation. Height of a wave.
removed incorrect tag, texified some formatting