Engineering Fundamentals: An Introduction to Engineering, 4th ed.c

616 Chapter 18 Mathematics in Engineering

(18.30)

In Equation (18.30), [A]

 1

is called the inverse of [A]. Only a square and nonsingular matrix

has an inverse. In the previous section, we explained the Gauss elimination method that you can

use to obtain solutions to a set of linear equations. Matrix inversion allows for yet another way

of solving for the solutions of a set of linear equations. As you will learn later in your math and

engineering classes, there are a number of ways to compute the inverse of a matrix.

18.6 Calculus

Calculus commonly is divided into two broad areas: differential and integral calculus. In the

following sections, we will explain some key concepts related to differential and integral calculus.

Differential Calculus

A good understanding of differential calculus is necessary to determine therate of changein

engineering problems. The rate of change refers to how a dependent variable changes with

respect to an independent variable. Let’s imagine that on a nice day, you decided to go for a ride.

You get into your car and turn the engine on, and you start on your way for a nice drive. Once

you are cruising at a constant speed and enjoying the scenery, your engineering curiosity kicks

in and you ask yourself, how has the speed of my car been changing? In other words, you are

interested in knowing thetime rate of changeof speed, or the tangential acceleration of the car.

As defined above, the rate of change shows how one variable changes with respect to

another variable. In this example, speed is thedependent variableand time is theindependent

variable. The speed is called thedependent variable, because the speed of the car is a function of

time. On the other hand, the time variable is not dependent on the speed, and hence, it is

called an independent variable. If you could define a function that closely described the speed

in terms of time, then you woulddifferentiatethe function to obtain the acceleration. Related

to the example above, there are many other questions that you could have asked:

What is the time rate of fuel consumption (gallons per hour)?

What is the distance rate of fuel consumption ( miles per gallon)?

What is the time rate of change of your position with respect to a known location (i.e., speed

of the car)?

Engineers calculate the rate of change of variables to design products and services. The engineers

who designed your car had to have a good grasp of the concept of rate of change in order to build

a car with a predictable behavior. For example, manufacturers of cars make certain information

available, such as miles per gallon for city or highway driving conditions. Additional familiar

examples dealing with rates of change of variables include:

How does the temperature of the oven change with time, after it is turned on?

How does the temperature of a soft drink change over time after it is placed in a refrigerator?

Again, the engineers who designed the oven and refrigerator understood the rate of change con-

cept to design a product that functions according to established specifications. Traffic flow and

product movement on assembly lines are other examples where a detailed knowledge of the

rate of change of variables are sought.

3 A 4

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3 A 4  3 A 43 A 4

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