Differentials

The Total Differential

In single variable calculus, we discussed the idea of the differential.  We now define the differential for functions of two variables.

 Definition Let z = f(x,y) then the total differential for z is Example

For

z  =  f(x,y)  =  x2 + xy

the definition produces

dz  =  (2x + y) dx + x dy

Exercise

Find dz if

z  =  x siny cosy

Application

In Chemistry we learn that

PV  =  NRT

where NR is a constant.  We can write:

PV
T  =                 =  k PV
NR

Suppose at 25 degrees C, gas is in an expanding cylinder of 55cc at a pressure of 3 atm.  Also suppose that the pressure is increased by 0.1 atm and the volume is decreased by 0.05 cc.  Then

dT  =  kPdV + kVdP

@  k [(3)(-0.05) + 50(0.1))  =  0.35k

so that there is an increase on the temperature of about k(0.35) degrees C

Suppose that you measured the dimensions of a tin can to be

h  =  6 0.1 inch

and

r  =  2 0.05 inch

What is the approximate error in your measurement for the volume of the can?

Solution

We have

V  =  pr2 h

Hence the error can be approximated by

D@  2prh Dr + pr2 Dh  =  2p(2)(6)(0.5) +p(4)(.1)

= 1.2 p + .4 p = 1.6 p @ 5.0 cu inches

Hence the volume is

V  =  75.4 5 cu inches

Differentiability

 Theorem:  If a function f(x,y) is differentiable at a point (a,b) then it is continuous at (a,b) where differentiable means           Df(x,y) = fx(a,b)Dx + fy(a,b)Dy + e1Dx + e2Dy where both e1 and e2 approach 0 as Dx and Dy approach 0.  Furthermore if the partial derivatives are continuous then the function is differentiable.

z  =  f(x,y)

is a differentiable function at the origin of two variables then

z = Ax + By + error

where the error term is small near the origin.  In other words the graph is approximately equal to a plane near the origin.