Exact Differentials
You know that the derivative of a function D 0 B C is just the 2 1
matrix `0 , `0 but suppose we are given cT B C UB Cd is there any
`B `C
function J B C so that .J cT B C UB Cd Yes, provided
`U
`T
`C `B Such a derivative is called an
Worldwide
Multivariable
Calculus
David B. Massey
in memory of my father, Robert Brian Massey (1934-2012), who taught me to love all
things mathematical and scientific
c 2012-2016, Worldwide Center of Mathematics, LLC
v. 0517160752
ISBN 978-0-9842071-3-8
C
Exact Differential Equations
The level curves J B C G of a function of two variables are a lot like
the solutions to a differential equation: they fill up the plane and no two
ever cross. In fact they are solutions to some differential equation. We're
goi
Souped up chain rule
When we differentiatiate a complicated function like sinB# " we thought of it as a
combination of two functions 0 ? sin ? and ?B B# " Then we differentiate
.
.
(sin ? cos ?
?B #B and multiply the results together
.?
.B
.
sinB# " = co
Directional Derivatives
`0
`0
and
give the slope of surface D 0 B C when we move
`B
`C
parallel to the B or C axis , but what if we move in some other direction? These slopes are
given by the directional derivative.
The directional derivative of the funct
Green's Theorem
So far in our study of multivariable functions we have covered two different forms of
integration, Line Intgrals and Double Integrals. Green's theorem ties these together.
If < is a simple closed curve and V is the region bounded by < then
Gradient
`0
`0
and
give us the slope of the
`B
`C
surface D 0 B C when we move on the surface in a direction parallel to
the B and C axes respectively. And the directional derivative .Z 0 gives us
the slope in the direction of the vector Z . Now we ask wh
Partial Fractions
Partial Fractions is a trick for integrating fractions like
Its actually just algebra razzle-dazzle and doesnt involve any calculus you dont already
know. The trick is to notice that the denominator looks like the common denominator for
Parametric
Parametric Equations of Curves
A parametric equations of a curve is just a vector function of a single variable:
B>
Z >
The variable > is called the parameter. We have already met these guys
C>
before as solutions to a system of differential
Line
Line Integrals
The Line Integral of a differential form [ B C cT B C U B C d would be better called the
B>
curve integral because it is the integral with respect to a parametrized curve <>
.
C>
Misnomer or not we write it ( cT B C U B C d.< Where b
Pathways
To get a straight-line path from the point + , to - . write
+
+
+
<> > " > > > ! to > "
,
.
,
,
.
example: a straight line path from (1,3) to (4,2)
"
%
" $>
<> " > >
> ! to > "
$
#
$>
For a path that goes along the curve C 0 B from B + to B ,