Chapter 1: Key Commands in Mathematica

For engineers and scientists, the 'Diff' and 'Int' commands in Mathematica are invaluable. As their names imply, 'Diff[…]' and 'Int[…]' facilitate the processes of differentiation and integration of functions, respectively. These fundamental calculus operations play a vital role in formulating problems related to the physical world, particularly through differential and integral equations, as we will explore in this post and those to follow.

Section 1.1: Understanding Derivatives

In Post #2, we examined the derivative of a function 'f' in relation to the variable 'x' using the 'D[f,x]' command.

Here, we can see the geometric significance of derivatives.

Subsection 1.1.1: Applications of Derivatives

Application #1: Finding Extremes of a Function

To identify the maximum or minimum values of a function of 'x', one must solve for the real roots of the equation formed by setting its derivative equal to zero. If the second derivative is negative at this root, the function has a maximum there (indicating a 'concave down' curvature). Conversely, a positive second derivative signifies a minimum. Consider the two straightforward examples below:

Application #2: Solving a Geometry Problem

Problem Statement: Determine the points on the circle defined by (x^2 + y^2 - 2x + 4y = 0) that are closest to and farthest from the point (P(4,4)).

To tackle this, we first plot the circle and the given point. Next, we find the coordinates on the circle before determining the point along the distance function of 'x' where the slope equals zero. Finally, we compute the coordinates on the circle that are nearest to and farthest from the point (4,4). Below are the steps carried out in Mathematica:

Section 1.2: Integration of Functions

If (F(x)) represents the integral of a function (f(x)), then the derivative of (F(x)) yields (f(x)). Consequently, an integral can also be termed as the anti-derivative of the same function (f(x)). Below, we present a few examples of integrals:

Chapter 2: Applications of Integrals in Geometry and Physics

There are numerous applications for integrals across both geometry and physics. Below, we illustrate a couple of examples related to geometry, and subsequently, physics. These applications typically employ either the 'Intersecting points by Solve' or 'FindRoot' commands alongside the 'Integrate' command.

Application #3: Calculating Area Between Curves

This classic problem is extensively utilized to compute the area enclosed by curves using their intersecting points as limits for integration. An example is shown below, highlighting 'Application #2: Geometry Problems involving these curves.' This application involves identifying the intersection points using 'Solve' and 'FindRoot', combined with the 'Integrate' commands in Mathematica.

Chapter 3: Ordinary Differential Equations

Engineers and scientists frequently encounter practical problems, both static and dynamic, modeled by mathematical differential equations (DE). These DE can be categorized as (1) Ordinary DE (ODE), where the unknown function depends on a single variable, and (2) Partial DE (PDE), where it depends on two or more variables. In this chapter, we will focus on ODEs and their solutions, while PDEs will be addressed in a future post. Below are examples of solutions to ODEs obtained using the 'DSolve[ ]' command in Mathematica:

Applications of ODEs in Physics

Many ODE problems arise in physics and engineering. For brevity, only one illustrative problem is outlined here:

Final Thoughts

The examples and applications discussed are merely a glimpse into the vast potential of Mathematica in solving and graphing solutions for real-world problems. Future posts will delve into various application problems that Mathematica can address effectively.

Next Post: Before we explore additional significant applications in engineering and science, we will discuss the fundamentals of vectors and matrices, including the operations associated with them.