Description for any change over time is necessary for most daily application of mathematics. Mathematics itself is applied everyday in many fields of expertise. Since each field of expertise have different requirement for the representation of change, mathematics itself have a lot of tools in its disposal to describe change.
Vector function over time
The easiest representation of change in mathematics is vector function over time. This representation of change is often used by students when they are studying basic physics. For example, when students are asked to determine the distance an object can reach if they are thrown with certain initial velocity and vertical degree, they are going to start by writing down the relation between the coordinate of said object in respect with the time elapsed. This relation is called vector function over time. By using this function, students could determine when vertical component of the position vector reach zero and calculate the horizontal component of the position vector at the same time.
In computer aided animation, vector function over time is often used to represent how a point in a vector graphic should change its position over time. Usually the representation of change itself is embedded inside the animation software itself. All the animator should do is to specify the position of a point for different time frame. The animation software could handle the linear algebra work automatically.
In the study of ballistic, vector function over time could be used to predict the position of a projectile given the time after the projectile was launched and where the projectile is going to hit the target.
Markov Matrix
Markov Matrix represents how a system is going to change their state for each previously defined time-step. The representation takes the form of a matrix and the change itself is calculated by the mean of matrix matrix multiplication. Usually the system to be studied have one or more vector representing its current state.
Markov matrix is widely used to predict how many times an activity should be done to convert something into something else. For example we could divide 100 students into 3 groups: 20 of them are A-getters, 50 of them are B-getters and the rest are C-getters. Past experiments show that giving students 1 hour extra lesson is going to turn 60% of C-getters into B-getter, 20% of C-getters into A-getters and 40% of B-getters into A-getters. By using this data we could predict that one hour extra lesson is going to result in: 46 students get A in their exams, 48 students get B in their exams and 6 students get C in their exams.
In finance, Markov Matrix could also be used to determine the probability of getting a possible investment result given the history of an investing instrument. By calculating the mean and standard deviation of interest rates from previous data, statisticians could determine the probability of getting an interest rate in the future. Then statistician could use the interest probability to calculate how likely it is to gain a possible of investment result.
Differential Equation
Differential Equations could be used to represent the amount of change of a variable in a system as a function of the current state of the system. Some differential equations could be solved analytically thus allowing us to predict the state of the system for any value to t, while the others could only be solved numerically.
Differential Equations are widely used in astronomy to represent the gravitational interaction between several objects in a planetary system. Given the mass, initial position vectors and initial velocity vectors of all objects within the planetary system, the differential equation could be used to predicts the positions and velocity of those objects in the future. By using the differential equations, astronomers could also prove whether the planetary system are going to stable or unstable.
In ecology, a differential equation system known as Lotka-Volterra model is also used to study the population of both predator and prey. By graphing the result of Lotka-Volterra integration, ecologist could study the dynamics of wildlife population in the wild.
Conclusion
Mathematics have various tools at its disposal to describe change. These descriptions of change enable mathematicians and scientist to make various kind of simulations. Usually the cost of using these simulations is cheaper than doing an actual experiment. These simulations had helped humanity to enhance our understanding on how nature works, by lowering the cost required to do many kind of experiments.
