EMAT10100 - Eng Maths 1Differentiation and Integration

John Terry2.10 QB

Lecture 7 - More Partial Differentiation

These were the DIFFERENTIAL and the TOTAL DIFFERENTIAL

In some sense, the differential represents the actual change,whereas the total differential is the approximate change.

In the last lecture we introduced two concepts for helping us toinvestigate the effect of small changes in individual variableson the rate of change of a total function.

How good an approximation does the total differential provide?

Example:

Consider the function z = xy. The definition of the totaldifferential tells us:

dz = y dx + x dy

What about the actual change?

When changes in x and y are governed by increments ∆x and ∆y respectively, we obtain the overall change ∆z as:

We may visualise this geometrically

Note that the total differential and the actual difference alter bya rectangle given by ∆x∆y

Example

Compute the total differential of:

The total differential is given by:

Answer

ExampleThe power (in watts) consumed in an electrical resistor isgiven by:

!

P =E2

R

If E=200V and R=8ohms, how much will the power alter byif E is decreased by 5V and R by 0.2Ω?

In this case we have:

Hence the total differential is given by:

Given E=200 and R=8, the amount varied is given by dE = -5and dR = -0.2.

Hence the power is reduced by approximately 125Watts

The total differential of a function is often used to find the degreeof accuracy of a quantity when measurements are affected bynoise and error.

Example

The volume of a cylinder of radius r and height his given by:

Measurements of the distances have shown that:

Find the greatest possible error in the calculation of V.

Example

The volume of a cylinder of radius r and height his given by:

Measurements of the distances have shown that:

Find the greatest possible error in the calculation of V.

Answer

Example: (Application of Total Differentials)

Suppose that:

where a and b have been determined experimentally withrelative error bounds given by p and q respectively.

Find the relative error bound on y.

‘relative error’ means:

< qWe are told:

How do we relate all these things together?

We will assume that the measuring equipment is good andtherefore the relative error bounds p and q are small.

In this case we may approximate

It then follows that

TOTAL DIFFERENTIAL

Differentiating y with respect to a and b we obtain:

Now we divide both sides by

<p <q

Hence

Exact Differential

We now discuss a relation of the total differential, theExact Differential.

Suppose that either through experimental observations ortheoretical considerations, we have an expression of the form:

The aim is to find a function f(x,y) (if it exists) such that:

What does all the mean?

Such a function f is termed the POTENTIAL FUNCTION andthe LHS of the expression:

is termed an EXACT DIFFERENTIAL

ExampleAssume that from an experiment we have the following:

If f(x,y) exists, then the following must hold:

This implies that:

Recall from last time that if f(x,y) is a continuous function then:

This provides a test for the existence of f

In our case we have:

How do we actually find our function f(x,y)?

Well

We can integrate both sides with respect to x (note that y behaves as if a constant) achieving:

where C is the ‘constant’ of integration.

Hence we actually have:

To find g(y) we use knowledge of the second partial derivative:

(*)

Differentiating (*) with respect to y we obtain:

Comparing the RHS of the two forms of fy we obtain:

Integrating:

Finally we combine these two parts to obtain f(x,y):

The POTENTIAL FUNCTION!

We now summarise the steps needed to determine if we havean EXACT DIFFERENTIAL:

1. Are the mixed derivatives the same?

yes no

2. Integrate fx to obtain possible form of f(x,y):f(x,y) = h(x,y) + g(y)

3. Differentiate the obtained function with respect to y andcompare with original form.

4. Integrate g’(y) and combine with f(x,y) obtained in step 2.to give the POTENTIAL FUNCTION

HOMEWORK

Read James pg. 641-647 Exercises 65,69,70 pg. 641Exercises 78,80,82 and 84 pg. 644Exercises 86,88 on pg. 647

Taylor Series in Higher Dimensions (James pg. 647-650)

We now discuss briefly the use of power series expansions offunctions of two variables.

Recall how a function f(x) could be expanded about x=x0 as:

In two dimensions we expand about a point (x0,y0) as follows:

Example: Expand f(x,y) = sin (xy) about the point