Skip to main content

Index Notation Partials and Rotations, a Cool Gradient Trick: EMII Notes 2014_08_09 Part I

Summary of what's gone on before.  Finally got the Levi-Civita to Kronecker delta identity down yesterday, 2014/08/08.  Today we're making more use of the index notation.  There are however, some notational stumbling points

$r^2 = x^2 + y^2 + z^2$

$\vec{r} = \left(x, y, z\right)$

Now if we want to take the derivative of both sides of the magnitude equation above, first remember we can write $r^2$ as

$r^2 = x_jx_j$

Now, finally taking the derivative of both sides o the above we get

$2r \partial_i r = 2x_j\partial_i x_j$

remembering the partial differentiation rules and the kronecker delta we can write the above down as

$2r \partial_i r = 2x_j \delta_{ij} = 2x_i$

which finally gives:

$\partial_i r = \dfrac{x_i}{r}$

The Kronecker trick above is crucial. Also remember, not one of the r's is a vector, they're all the magnitude of the vector.



\section{Rotations}

Any rigid rotation of a vector can be defined as:

$\begin{pmatrix}
x'\\
y'\\
z'
\end{pmatrix} = M\begin{pmatrix}
x'\\
y'\\
z'
\end{pmatrix} $

Here are a few of the important parts.  The Matrix $M$ is orthogonal and, $M^TM = 1$

The transpose actually defines orthogonality.  If such a matrix is of dimension n, then it is n $)\left(n\right)$ matrix.

The footnote has all the cool kid stuff about rotation matrices and how to name them:

First of all, if the determinant of a matrix is +1, then it's a special orthogonal matrix, $SO\left(n\right)$.  The other sort, the sort with a determinant of -1 are actually rotations with a reflection of the coordinates.

There are also some identities in the footnote that will come in handy

$det\left(AB\right) = \left(det A\right)\left(B\right)$

and

$det\left(A^T\right) = det\left(A\right)$

Using the above two identities, we can see that $\left(det M\right)^2 = 1$

Memorize these!!!

In index notation, the rotation above can be written as
$x_i^{\prime} = M_{ij}x_j$

The orthogonality condition becomes

$M_{ki}M_{kj} = \delta_{ij}$







Comments

Popular posts from this blog

Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla



to the nabla for another coordinate system, say… cylindrical coordinates.



What we’ll need:

1. The Cartesian Nabla:



2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:



3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:



How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…

The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1.  While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero.  This is typically explained in the following way.  While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare.  That doesn't stop us from looking for them though!

Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2].  Cabrera was utilizing a loop type of magnetic monopole detector.  Its operation is in concept very simpl…

Kids R Kapable

Just a little note to concerned ‘grownups’ everywhere.  If you look at a kid—and I mean really look—I don’t mean notice a person shorter than you, I mean make eye contact, notice their facial expression and observe their body language—If you look at a kid, don’t assume they need your help unless they’re obviously distressed, or ask for it.  You might think this is difficult call to make.  You might think, not having kids of your own, that you’re unable to make this determination.  You are.  You do in fact, already have the skills even if you’ve never been around kids  It’s a remarkably simple call to make, just use the exact same criteria you would for determining if an adult was in distress.  Because, guess what, kids and adults are in fact the same species of animal and communicate in the same way.  Honest.  If someone—adult or child—doesn’t need your help, feel free to say hello, give a wave, give a smile, but don’t—do not—try to force help on anyone that doesn’t want or need it.



Y…