Physics 225
9.3
9.3
Unit 9: Differential Calculus … in 3D
We’re done with Special Relativity and we’re moving on.
Our overarching goal for the second half
of the course is to study
electromagnetism
in the way it was meant to be studied: using
3D differential calculus
.
Along the way, we will encounter many mathematical tools, introducing
them always in a physical context.
Our destination is to see
Maxwell’s equations
in their most
powerful form —
differential form
— and learn how to use them.
When we reach our goal, I
promise you: you will be so familiar with differential calculus that you will be able to read
these
beautiful equations like English sentences.
Exercise 9.1:
Fields
Death to jargon!
“Jargon” means “a technical term that is used to conveniently
represent an idea in some specific area of expertise”.
To experts in that area, jargon is great: everyone knows what
it means, so you can use one word of jargon to convey a
complex idea that would otherwise take a whole sentence to
explain.
But
until
you become an expert,
jargon is miserable
and can be
very dangerous!
When an
expert-sounding person permeates their speech with jargon whose meaning is unclear to you, you
must ask
.
“What does word X mean?”
All too frequently, people (not just students!) are
intimidated by jargon.
People are generally scared of sounding ignorant, so they don’t ask what
word X actually means because they think it’s supposed to be obvious.
Pretty soon, they’ve heard
word X enough to figure out roughly what it means, so they start constructing sentences using word
X (basically by copying what they’ve heard other people say).
Before long, this becomes a terrible,
terrible
habit.
You can find yourself far down the road of your studies and routinely using terms
whose meaning was
never
clear to you in the first place.
Building your knowledge on jargon you
don’t really understand is like building a house on sand: it gets shakier and shakier as it grows, and
that is one very uncomfortable house to live in.
You know what?
Most of the time, the fancy-sounding jargon that makes folks sound so expert and
seems so intimidating isn’t complicated at all
.
You just have to ask … and ask early!
Here is our first piece of Jargon Assassination for today:
fields
.
Electric
fields, force fields, gravitational fields, magnetic fields … Fields start
popping up all over the place in 212 and they sound like something very
deep and profound.
We’ll they’re not.
A
field
is a
function
that depends on
more than one spatial coordinate
,
That’s it: a field is a function in 2D or 3D space.
f
(
x
) is an ordinary 1-
dimensional function, while
f
(
x
,
y
) is a field in 2D space, and
f
(
x
,
y
,
z
) is a
field in 3D space.
A field is very much like a
map
: it’s a map of some
interesting quantity within a region of 2D or 3D space.