This question just recently came up in a physics problem, but it's actually more of a math problem, so I'm going to ask it here.(adsbygoogle = window.adsbygoogle || []).push({});

Let me provide some context.

The HW was to show mathematically that the divergence of the curl was zero for the first part, and to explain it physically/conceptually.

The assignments done, I'm not asking for help with that, but...

I was thinking, the determinant grants an nth dimentional volume of the parallelipiped created by the vectors containing it, and this gets really interesting when you consider how we calculate curl (and cross products) with a determinant.

Let's have a vector field in R^3 ##F:=<F_x,F_y,F_z>## or ## F_x \hat{i} +F_y \hat{j} + F_z \hat{k}##

and we want to calculate the curl.

##\left | \begin{array} c

\hat{i} &\hat{j} & \hat{k} \\

\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\

F_x & F_y & F_z \\

\end{array} \right |##

Now the bottom line is just our vector field, expressed as a row vector, but each entry is the magnitude of each component. In other words, each entry (by itself) is a scalar.

The middle row is our vector operator del, expressed as a row vector, but, again, each entry is the magnitude of each component. Each entry is also a scalar. The magnitude of each component of our bottomm two row vectors is a scalar, as expected, and also (to me) as makes sense.

Now we get to that pesky top row. i j k hat, seems harmless enough, but think about it. What that row is saying, is that in our vector-space R^3, there exists a vector such that the magnitude of the i hat component is i hat (not 1, i hat), the magnitude of the j hat component is j hat (again, j hat, not 1), and the magnitude of the z component is k hat!

This to me seems bothersome. There is a 3x3 vector array within that top row vector, in otherwords, we have a vector such that:

##v :=

\left [ \begin{array} c

1 & 0 & 0 \\

0 & 1 & 0 \\

0 & 0 & 1 \\

\end{array} \right ] ##

maybe I didn't express that last vector array correctly, but the first column (or row for that matter, it's all the same in this case) is the x component of our vector, the second column is our y component, and the third is the z. The components of our vector are in fact vectors.

So if I am to interpret this physically, it starts to get really abstract. First off, if we have 3 vectors, just normal vectors in R^3, and we calculate the determinant of their matrix, we get a volum, but this volume doesn't exist in R^3, this is a volume of the vector space that exists in R^3 (the vector corresponding to a field, for example, doesn't actually extend into physical space, that vector only represents the field at a point, but we're using it's magnitude and direction to gather the volume generated within this vector space).

So that's a little bit abstract, but not too bad. Now when we add in our vector array, the only way I can see to interpret this, is that this vector, itself, contains within iteself a vector space. This is not a direct subspace however, this is a dimension within a dimension. On top of that, it's a spatial dimention within a spacial dimension! If that wasn't enough, it's actually 3 spacial dimensions within 1 spacial dimension. So in calculating the curl of a field, are we actually calculating the volume of some messued up vector space which has "tiny dimensions" within itself? If it is, is this what's meant by the tiny dimensions that you hear in reference to string theory? (That's a completely off topic and random question that popped up when I wrote that line, I'd like to hear an answer, but not to get bogged down on it).

So, the curl(/cross product): the volume between a 2 vectors in R^3 and a vector space that exists within the dimensions of R^3? Seems a little messed up to me, but hopefully I can be enlightened.

Ok also, I used to start my arrays with \left [ \begin{array} ccc for a 3x3, cc for a 2x2... etc, but that's giving me some random c's in my array now? What is that c actually for? I figured it was the number of columns, but I guess not?

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# A vector whose components are vectors?

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