The operator j and complex numbers.

[mark_in_manc]

Originally Posted by Radio Wrangler

The format in which Maxwell's equations were derived.

You got me; I wasn't an elec eng grad!

[Craig Sawyers]

At the risk of exp-anding the topic, but still keeping in the remit of circuit analysis and modelling, how about the Kronecker delta function, the Dirac delta function and Heavyside step function? At least the definition of these animals doesn't involve complex numbers.

[Herald1360]

Originally Posted by Craig Sawyers

At the risk of exp-anding the topic, but still keeping in the remit of circuit analysis and modelling, how about the Heavyside step function?

Would that be related to the title sequence in Monty Python, perhaps?

[Radio Wrangler]

Maxwell was at Cambridge studying mathematics where he was introduced to quaternions.

When he developed his field equations, he worked in terms of quaternions and first published them in quaternion form. Only later did they get translated to the form known today. I suspect they're the main reason why grads, divs and curls still get taught.

quaternion sounds like the name of one of those fictitious particle that star trek script writers keep having to invoke to rescue themselves after they've written themselves into a corner "Scotty modified the transporter to emit a beam of quaternions to scare the revolting students away from the ship..." It would work!

David

[dsergeant]

Never heard of quaternions but div, curl and grad (in that order, not yours) I well remember as the concept at University I could just not get my head round. I nearly failed the electromagnetics paper in my second year exams when many of us went down with food poisoning on the day (steak and kidney pie at hall went a bit wrong) - the Maxwell questions like the food were largely left untouched.

Dave

[Craig Sawyers]

Way back I did an Electronics degree, but chose as many theoretical and mathematical options as I could because I enjoyed it. I'd have to bone up a bit to refresh my brain cell, but I was pretty comfortable with all that stuff (complex numbers, functions of a complex variable, vectors and vector fields, EM theory, solid state theory, partial diff equations etc etc).

But quaternions? No thanks very much - they are totally impenetrable. The other thing I could not get my head around at all was Green's Functions, which is still a frustration to me to this day.

[G8HQP Dave]

I too struggled with Green's functions. I felt that somehow I had slept through the lecture which introduced them, and so missed some vital piece of information which would somehow make them clear. Just as likely is that they never gave us this information. There were several occasions in which the Nth year lecturer said "you will do this properly next year, so I will skip the details" then in the N+1th year someone else said "you saw this last year, so I will just use the result without proof". At the time I thought they were just being a bit lazy, but now I wonder if it was because they struggled with the maths too?

I heard somewhere that quarternions have found a use in analysing the movements of robot arms so they don't clash.

[julie_m]

Quaternions sound nice and scary, what with being impossible to visualise properly since they exist as vectors in four dimensions, and most people's brains only have room to fit three .....

Am I right in thinking:

1. i * i = j * j = k * k = -1
2. i * j * k = -1
3. i * j = j * i = k
4. i * k = k * i = j
5. j * k = k * j = i

? And from then on, it's just like ordinary complex numbers, you have to keep the real, i, j and k terms separate?

[kalee20]

Nearly.

Multiplication of quaternions is partly anticommutative,

So while i*j=k, swap them around and you get j*i=-k. Etc!

I can well imagine that quaternions are used to analyse swinging arms, because they do have applicability to spherical geometry. But I never studied that!

[julie_m]

Wow, order-dependent multiplication ..... Now that's some seriously weirdy maths going on there!

So does i * j = k but j * i = -k, j * k = i but k * j = -i and k * i = j but i * k = -j ? Or have I got one or more of the minus signs wrong?

[Radio Wrangler]

It's mostly a vector thing. You can define i,j,k in either a left-handed order or in a right handed order, then you get into vector system representation of material E-M properties and before you know where you are you've got a microwave circulator!

David

[G8HQP Dave]

Wow, order-dependent multiplication ..... Now that's some seriously weirdy maths going on there!

Such things are commonplace in quantum mechanics - in fact it could be said that this marks one of the fundamental differences between QM and the classical world.

[kalee20]

Originally Posted by julie_m

Wow, order-dependent multiplication ..... Now that's some seriously weirdy maths going on there!

So does i * j = k but j * i = -k, j * k = i but k * j = -i and k * i = j but i * k = -j ? Or have I got one or more of the minus signs wrong?

No, that's right.

Really, you only need your original two 'rules' to work everything out:

1) i*i = j*j = k*k = -1
2) i*j*k = -1

Because, if you left-multiply (2) by i, you get:

i*i*j*k = i*(-1)
giving (-1)*j*k = -i or equivalently, j*k = i.

And if you right-multiply (2) by k you get:

i*j*k*k = (-1)*k
giving i*j*(-1) = -k or equivalently, i*j = k.

But if you right-multiply this last result by j, you get:

i*j*j = k*j
giving i*(-1) = k*j, or k*j = -i. Etc etc etc!

Reversal of signs does occur between multiplying between any two different 'imaginary' quantities, but doesn't between an imaginary quantity and 1 or -1. It has the interesting result that many equations have loads of answers: the equation z² + 1 = 0 has an infinite number of solutions with quaternions, whereas in complex numbers there's just two, i and -i.