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Differential calculus refresher

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One of the first mathematical tools a neophyte engineer learns is calculus. The basics of limits, differentiation, and integration permeate all of engineering mathematics.

Many of the mathematical tools engineers use to evaluate and predict behavior, such as vibrations, require equations that have continuously varying terms. Often, there are many terms regarding the rate of change, or the rate of change of the rate of change, and so forth, with respect to some basis. For example, a velocity is the rate of change of distance with respect to time. Acceleration is the rate of change of the velocity, which makes it the rate of change of the rate of change of distance with respect to time. Determining the solutions to these types of equations is the basis of differential calculus.

An equation with continuously varying terms is a differential equation. If only one basis is changing, then it is an ordinary differential equation (ODE); however, if two or more bases are changing, then it is a partial differential equation (PDE). An ODE uses the notation "d" and a PDE uses ∂ to refer to change.

Understanding differentiation starts with an understanding of limits.

Limits

A graph is a useful method for determining how an equation behaves. The independent variable t in Eq. 1 determines how the dependent variable y behaves. The operators and constants in an equation specify this behavior. Fig. 1 shows the graph of Eq. 1, the distance of freefall over time with an initial velocity of zero. Down is considered negative in this equation:

Vol1 page 0001 eq 001.PNG....................(1)

The x-axis (abscissa) usually is the independent variable, and the y-axis (ordinate) usually is the dependent variable; however, many drilling charts hold an exception to this generality, in that their ordinate often is the independent variable, and their abscissa is the dependent one. An example of such a drilling chart is the depth vs. time graph.

  • Fig. 1 – Graph of Eq. 1, the equation of free fall on earth.

    Fig. 1 – Graph of Eq. 1, the equation of free fall on earth.

In Fig. 1, at the time of 3 seconds, the distance is –96.522 ft. A tangent line to the graph at 3 seconds is known as the slope (A) of the graph at that point. To quickly estimate the slope of the tangent, divide the rise (Δy) by the run (Δt), as shown in Eq. 2:

Vol1 page 0002 eq 001.PNG....................(2)

In this case, the tangent y value at 2 seconds is –48.261 ft and at 4 seconds is –241.305 ft. The slope then is:

Vol1 page 0002 eq 002.PNG....................(3)

Because the units in this case are ft/sec, this slope gives the velocity at that point. It is the rate of change of the distance with respect to time.

A limit is defined as the value of a function at a given point as that point is approached from either higher or lower values (often referred to as approaching from the left or right, respectively). The limit (Y) of Eq. 1 at 3 seconds is:

Vol1 page 0002 eq 003.PNG....................(4)

Y is known as the limit of the function. In this simple case, Y is the same regardless of whether t approaches 3 from the left or the right. This is not true in all cases, however (e.g., with a discontinuous function). In these cases, the limit can be determined analytically. One can also determine the limit using a graph such as in Fig. 1.

Limits have the following properties:

Vol1 page 0002 eq 004.PNG....................(5)

Vol1 page 0002 eq 005.PNG....................(6)

Vol1 page 0003 eq 001.PNG....................(7)

and

Vol1 page 0003 eq 002.PNG....................(8)

Derivatives

As noted earlier, the slope of graph of Eq. 1 at 3 seconds = –96.522 ft/sec and is the velocity (v) of free-fall at 3 seconds from release. This value is known as the first derivative of Eq. 1 at the value of 3. It is written as:

Vol1 page 0003 eq 003.PNG....................(9)

and is defined as:

Vol1 page 0003 eq 004.PNG....................(10)

As the limit of the value of Δt approaches zero, the solution converges to the first derivative.

Derivatives have the following properties (r = constant).

Vol1 page 0003 eq 005.PNG....................(11)

Vol1 page 0003 eq 006.PNG....................(12)

Vol1 page 0003 eq 007.PNG....................(13)

Vol1 page 0003 eq 008.PNG....................(14)

Vol1 page 0003 eq 009.PNG....................(15)

Vol1 page 0003 eq 010.PNG....................(16)

In the case of Eq. 7, where Q = 0, L’Hopital’s rule can help find the limit. This is shown in Eq. 17:

Vol1 page 0003 eq 011.PNG....................(17)

Other rules regarding differentials are the following.

The linear superposition rule:

Vol1 page 0004 eq 01.PNG....................(18)

The product rule:

Vol1 page 0004 eq 02.PNG....................(19)

The quotient rule:

Vol1 page 0004 eq 03.PNG....................(20)

The chain rule (or function of a function):

Vol1 page 0004 eq 04.PNG....................(21)

Multiple differentiations can be shown by

Vol1 page 0004 eq 05.PNG....................(22)

and continued differentiations can be shown by

Vol1 page 0004 eq 06.PNG....................(23)

A useful point to recognize is where a slope equals zero, which can correspond to a maximum, a minimum, or an inflection. To determine these points, determine a first derivative of an equation. Then, set this first-derivative equation to equal zero and solve for the basis (the unknown). To determine whether this point is a maximum, a minimum, or an inflection, determine the second derivative of that equation. If that value is negative, the point is a maximum; if it is positive, the point is a minimum; and if it is zero, the point is an inflection.

The graph of Eq. 24 ( Fig. 2 ) is an example of this process:

Vol1 page 0004 eq 07.PNG....................(24)

The first derivative of Eq. 24 is:

Vol1 page 0004 eq 08.PNG....................(25)

which, when set equal to zero, is a quadratic equation with two roots, t = 3 and 1/3 . These two points correspond to the maximum and minimum points on the graph. To prove which is which, a second derivative is taken:

Vol1 page 0005 eq 001.PNG....................(26)

which at t = 3 and 1/3 is equal to 8 and –8, respectively. This means that at t = 3, the function is at a minimum and at t = 1/3, the function is at a maximum.

  • Fig. 2 – Graph of Eq. 24, an example of slopes, minima, and maxima.

    Fig. 2 – Graph of Eq. 24, an example of slopes, minima, and maxima.

The first differentiation of the equation of the position of a free-falling object starting at rest (Eq. 1) gives the slope of the graph, which, as noted, is the velocity:

Vol1 page 0005 eq 002.PNG....................(27)

A second differentiation gives the change of the slope with respect to time (acceleration), and is:

Vol1 page 0005 eq 003.PNG....................(28)

which is the acceleration caused by Earth’s gravity.

Differential-equation solutions

Solutions to differential equations solved in closed form can range from trivial to impossible. Numerical methods often are required. Nevertheless, some general strategies have been developed to solve differential equations.[1][2][3]

An ODE with only first derivatives is known as a first-order ODE. A second-order ODE has second and possibly first derivatives. The same reasoning applies to third order and beyond. Likewise, when a PDE has only first derivatives, it is a first-order PDE. The second and third orders and beyond are defined on the basis of their highest-order derivative.

This section has covered some of the basics of ODE and PDE mathematics. The reader is urged to review mathematical texts and handbooks for more details on this subject.

Nomenclature

A = slope, dimensionless
gc = gravitational constant, L/t2, 32.174 ft/sec2
P = generic value
Q = generic value
r = generic constant
t = time, seconds
y = dependent variable, various
Δt = change in time, t, seconds
v = velocity, L/t, ft/sec
Δy = change in dependent variable, various

References

  1. ↑ Fanchi, J.R. 1997. Math Refresher for Scientists and Engineers. New York: John Wiley & Sons.
  2. ↑ Leithold, L. 1972. The Calculus with Analytic Geometry. New York: Harper and Row.
  3. ↑ Bird, J.O. 2001. Newnes Engineering Mathematics Pocket Book, third edition. Oxford, UK: Newnes.

Noteworthy papers in OnePetro

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External links

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See also

PEH:Mathematics_of_Vibrating_Systems

Category

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