Vectors: Grad, Div and Curl

Another chapter down!  This is my favorite part.  It's like someone is reading my mind.

MATH: Were you looking for this?
RYAN: Um.  I dunn-
                  Omigod.  ..... Ummmmm.  This is totally awesome.  Yes. Yes I-
MATH: There's more, if you'd --
RYAN:  ...was.   wait.  What?
MATH ... stop talking.
RYAN:   Oh.
MATH:  Check this out:

A scalar function φ is said to be harmonic if it is continuous,

 has continuous second partial derivatives, and satisfies Laplace's Equation:

      

RYAN:  How did you know I-
MATH: I'm just getting warmed up.  How about this.

RYAN: But how -?
MATH: ...do you define the gradient of a vector field?  Turn the page.
                    {rustle}

     Like that.  Now try this:

     And you have a formula for computing Del squared of a field:

     You should probably check the partials by hand.
RYAN:  Yeah, I-
MATH: Oh, and you're welcome.
RYAN:  Hey! I was go-
MATH: Sure you were.
RYAN:  This shit is fucking amazing.
MATH: {Yawn}Yeah.  {Stretch}  I know.  So, I see the Indian girl is still at the next table.  You know it's not because she wasn't going to drink it, right?
RYAN:  Wait. What?
MATH:  The beer.  You should talk to her. She's taking a long time to put her hair up for you.  You should probably ask her out.
RYAN:  Look, buddy. I'm not a mind reader.  If she wants to talk to me, I'm right here.
MATH:  Spare me.    And good luck with that.
RYAN:  With what?
MATH:  ....
            ....
 RYAN:  What?
MATH:  You are a complete idiot.

____________________

I am a complete idiot.  Anyway, Vectors continue to contain more Radness per component of space than any other container.  Expressed in terms of  BMX,

Where grad(BMX) is the direction of maximum increase in BMX, in mega-wheelie-hours.

I tend to double, triple, sextuple-check identities.  That has to stop.  The vector algebra identities involving Del take an Awful Lot of Symbols to verify by hand.    I did them all. A few results will live here so I can find them.  Like

curl (curl f) =

Notes:

1) The proof is just a verification of consistency:

If the operator distributes partials according to

       1) the usual product rules for vectors,

        2) swapping dot product order  to allow Del to operate on something

Then a result is obtained which is algebraically consistent.  In other words, it might as well be a definition.

I think it is self-evident that the intermediate, undefined form grad(f)  preserves measure if immediately operated on by dot (scalar) product.    It is not a proof that grad(f) has any general meaning.  

I assume the justification for adoption of the algebra itself lies in physical measurement of space. 

2)  I have almost found my difficulty with partial derivatives, implicit functions, and multiple integration.  The place where all of the differentials come together remains unclear.  There is more than one way to integrate multiple variables. It depends on the relationship between all quantities, there are many such quantities, and I cannot convince myself that substitution of a variable is trivial.  I think very soon I will have to go back and build diagrams with every little dx, dy, dz,  and work out each connection to a partial or substituted variable.

Grrrr.

In the meantime, moving forward is helping, so I will keep doing so.  On to Vector Integration.