13.4 Differentials

“Mathematics possesses not only truth, but supreme beauty - a beauty cold and austere, like that of sculpture."

-Bertrand Russell

Recall from Calc AB, Differentials:

When we first started to talk about derivatives, we said that

becomes when the change in x and change in

y become very small.

y

x

dy

dx

dy can be considered a very small change in y.

dx can be considered a very small change in x.

Let be a differentiable function.

The differential is an independent variable.

The differential is:

y f xdxdy dy f x dx

Example: Consider a circle of radius 10. If the radius increases by 0.1, approximately how much will the area change?

2A r

2 dA r dr

2 dA dr

rdx dx

very small change in A

very small change in r

2 10 0.1dA

2dA (approximate change in area)

2dA (approximate change in area)

Compare to actual change:

New area:

Old area:

210.1 102.01

210 100.00

2.01

.01

2.01

Error

Original Area

Error

Actual Answer.0049751 0.5%

0.01%.0001.01

100

Why do differentials work?

They work due to the concept of local linearity.

Differentials (in 2D) are equivalent to using a tangent line to make an

approximation

This definition can be extended to a function of 3 or more independent variables:

In more variables differentials work like this:

A diagram that shows approximation by differentials

Example 1

Find the total differential for the functions.

Evaluate the z function at (2,0) use the total differential to approximate the z function at (2.1,-.1)

Solution to example 1

Plug in values x=2, y= 0 and dx = .1 and dy = -.1

Example 4

If the potential error in measuring a box is .01 cm and the dimensions are found to be x = 50 cm, y = 20 cm and z = 15 cm. Use differentials to find the possible error in the volume of the box. Note: V = xyz

Solution to example 4If the potential error in measuring a box is .01 cm and the dimensions are found to be x = 50 cm, y = 20 cm and z =

15 cm. Use differentials to find the possible error in the volume of the box. Note: V = xyz

Example 2

Show that the function is differentiable at every point in the plane.

Example 2 method 1

There are no points of discontinuity for f(x,y)

fx = 2x which is continuous every where

fy = 3 which is continuous every where

Therefore f(x,y) is differentiable everywhere

Example 2: method 2 using the definition of differentiable

To demonstrate continuity you can either demonstrate

lim f(x,y) = f(a,b)(x,y)→(a.b)This is often hard because as you will recall proving that a limit exists for a function of 2 or more independent variables is usually challenging

OR

You can show that f(x,y) is differentiable at (a,b) by Thm 13.5 (which is often easier to do)

Example 5

Show that f(x,y) is not continuous (and there by not differentiable) at (0,0)

A graph of the function:

Example 5 solution

Solution: You can show that f is not differentiable at (0,0) by showing that it is notcontinuous a this point. To see that f is not continuous at (0,0), look at the valuesOf f(x,y) along two different approaches to (0,0).Along the line y=x, the limit is