8—Multivariable Calculus 185y
x
dy
dx
For two variables, the picture parallels this one. At a point(x,y,z) = (x,y,f(x,y))find the
planethat best approximates the function in the immediate neighborhood of that point. Set up a
new coordinate system with origin at this(x,y,z)and call the new coordinatesdx,dy, anddz. The
equation for a plane that passes through this origin isαdx+β dy+γ dz = 0, and for this best
approximating plane, the equation is nothing more than the equation for the differential, Eq. (8.13).
dz=
(
∂f(x,y)
∂x
)
ydx+
(
∂f(x,y)
∂y
)
xdy
dx
dy
dz
The picture is a bit harder to draw, but with a little practice you can do it.
For the case of three independent variables, I’ll leave the sketch to you.
Examples
The temperature on the surface of a heated disk is given to beT(r,φ) =T 0 +T 1
(
1 −r^2 /a^2
)
, whereais the radius of the disk andT 0 andT 1 are constants. If you start at positionx=c < a,y= 0and
move parallel to they-axis at speedv 0 what is the rate of change of temperature that you feel?
Use Eq. (8.4), and the relationr=
√
x^2 +y^2.
dT
dt
=
(
∂T
∂r
)
φdr
dt
+
(
∂T
∂φ
)
rdφ
dt
=
(
∂T
∂r
)
φ[(
∂r
∂x
)
ydx
dt
+
(
∂r
∂y
)
xdy
dt
]
=
(
− 2 T 1
r
a^2
)[ y
√
x^2 +y^2
v 0
]
=− 2 T 1
√
c^2 +v^20 t^2
a^2
. v
2√^0 t
c^2 +v^20 t^2
=− 2 T 1
v^20 t
a^2
As a check, the dimensions are correct (are they?). At time zero, this vanishes, and that’s what
you should expect because at the beginning of the motion you’re starting to move in the direction
perpendicular to the direction in which the temperature is changing. The farther you go, the more
nearly parallel to the direction of the radius you’re moving. If you are moving exactly parallel to the
radius, this time-derivative is easier to calculate; it’s then almost a problem in a single variable.
dT
dt
≈
dT
dr
dr
dt
≈− 2 T 1
r
a^2
v 0 ≈− 2 T 1
V 0 t
a^2
v 0
So the approximate and the exact calculation agree. In fact they agree so well that you should try to
find out if this is a lucky coincidence or if there some special aspect of the problem that you might have
seen from the beginning and that would have made the whole thing much simpler.