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1d
answered How are these definitions of the inertia tensor the same?
2d
comment Struggling with “technique-based” mathematics, can people relate to this? And what, if anything, can be done about it?
True, you can't; it's possible that there is a series of zeros that do not form a continuous set that nevertheless forbids any neighborhood not containing zeros. I have no idea what the solution would be in such a case.
Apr
14
answered Little mistake with Levi-Civita symbol property
Apr
9
comment Very confused about directional derivatives as vectors
I ask only because it's easy to run in circles answering such a question, but merely having to prove the uniqueness and existence of such a covector (rather than having to prove that it coincides with the exterior derivative of a coordinate function) is a clear and unambiguous task.
Apr
9
comment Very confused about directional derivatives as vectors
Then what do you wish to prove? That such a covector exists? Or that it coincides with the exterior derivative of the corresponding coordinate function? Or something else?
Apr
9
comment Very confused about directional derivatives as vectors
$\partial$ isn't something you should expect to cancel through division--it isn't something you're multiplying in the first place. - What's your definition of $dx^\mu$? The exterior derivative of a coordinate function? The unique covector such that $dx^\mu(\partial_\nu) = {\delta^\mu}_\nu$? Something else?
Apr
6
answered Intuition of Greens Theorem in the plane
Apr
5
comment Why is the argument of complex number determined up to integer multiple of $2 \pi$?
That's not what determined means here. Compare: the antiderivative of a function is determined up to a constant function. This does not mean the antiderivative is a multiple of a constant function.
Apr
4
comment How to prove that below quantity is a Third Rank Tensor
Is this in special relativity? In 4 dimensions?
Apr
3
answered on a particular expression of riemann curvature tensor
Apr
2
comment Definition of divergence operator
Of course. The integral form of divergence (or curl, or gradient) is very useful; if it's of interest to you, you can use the same procedure in spherical or cylindrical coordinates (instead of a perfect cube, you'll have to use pieces of spheres or cylinders), and this is convenient way to derive the divergence, gradient, and curl formulas for general coordinate systems.
Apr
2
comment Definition of divergence operator
That's basically what we're doing with the Taylor expansion--showing that any terms that vary over the surface must become small in the limit $L \to 0$.
Apr
2
comment Definition of divergence operator
Sorry, the notation is a little sloppy. Ultimately what you want to do is write $F_x, F_y, F_z$ each in terms of a Taylor expansion around the central point and in terms of the side length $L$. In turn, this turns the integrals over $x, y, z$ into integrals over $L$.
Apr
2
answered Definition of divergence operator
Apr
2
comment Definition of divergence operator
This is the surface of a cube, centered at the point at which you want to evaluate the divergence. If you know the $(x,y,z)$ coordinates of that point, you should be able to determine the bounds of each face. The length of each face is arbitrary (pick some length $\ell$) and should go to zero as $V$ goes to zero.
Apr
2
comment Definition of divergence operator
For $dA$, using the surface of a cube means it will only have one component on each face. For instance, the top face is $(0, 0, dx \, dy)$.
Apr
2
comment Definition of divergence operator
Yeah. If you were using $\nabla$ directly instead, wouldn't you end up writing something in terms of partial derivatives of those components? So those components seem like good things to start with.
Apr
2
comment Definition of divergence operator
Are you able to write the vector field $F$ and the surface element $dA$ in terms of their cartesian components and the cartesian basis vectors?
Apr
1
comment Definition of divergence operator
Have you considered, for example, the surface of a cube as the surface of integration?
Mar
31
comment Verifying Vector Operation Identities
What is your definition of $\nabla$ (or of curl, divergence, and gradient)?