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. 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:

Example: Consider a circle of radius 10. If the radius increases by 0.1, approximately how much will the area change? very small change in A very small change in r (approximate change in area)

Compare to actual change: New area: Old area:

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 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

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) f x = 2x which is continuous every where f y = 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 not continuous a this point. To see that f is not continuous at (0,0), look at the values Of f(x,y) along two different approaches to (0,0). Along the line y=x, the limit is