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Differential Equations and Calculus
Differential equation is perhaps the most important mathematical tool in the field of science and engineering.
All higher level mathematics is more or less dependent upon calculus and differential equation. Calculus is everywhere. Newton first applied it to solve problems related to real world. He used very small quantities to define differentiation and integration. This small vanishing quantities were known as the "ghost of departed quantities". His method worked but fundamentally it was flawed , which was later redefined as limiting value.
Greek used the method of exhaustion to find the area of a circle ,
which informally gave the idea of integral calculus. To find the area of a circle Greek first inscribed a polygon inside the circle approximating the area of the circle by that polygon. The polygon is of arbitrary numbers of sides. As the number of side s increase s to infinity the area of the polygon will give the eaxact area of the circle. Greek kne w that this worked but they did not know why this would be always the case. This method of finding area ultimately provided a great motivation to develop basics of integral calculus.
Limit of a function or sequence is defined to give rigorous foundation to calculus. What is limit is hard to explain but mathematics has a precise definition for it.
A quantity is said to be the limit of another quantity, when the quantity approaches any given quantity, however small, without the quantity approaching, so that the difference between the quantity and the limit is unassignable.
Or, if a given quantity converges to another quantity making the difference between them less than any arbitrary small number within an interval of certain time then in the end the two quantities ultimately become equal.
we can prove this lemma of Newton in the following way:
let us suppose the two quantities do not become equal. So there will be a certain numerical difference between them, which will not be less than any arbitrary small number. This is contrary to our hypotheses. That is, the two quantities must be equal.
limiting value of a function exists when the function is continuous. A function is continuous when sufficiently small change in input produces a small change in
the output of the function. Otherwise the function is discontinuous. When a graph of any function in the Cartesian plane is unbroken, the function constituting the
graph is said to be continuous in the domain of the function. The domain of a function can be a part of real line or entire real line.
Continuity of a function can also be defined using the limit. When a function is continuous , its limit exists at every point where the function is defined. Limit is a value of a function, which the function approaches when another variable approaches some value . This definition is not satisfactory anyway. Newton came up with an idea of limit but could not define it mathematically. Later limit was defined and calculus was dragged out of shaky foundation. We can define limit in the following way using an inequality :
Limit can be defined using class and relation. A quantity x is said to be a limit of class c with respect to relation P if 1) c has no maximum in P
2) every member of c in the field of P precedes x and 3) every member of the field of P which precedes x precedes some member of c . So the limit x may or may not belong to class c. This definition of limit gives all the essential properties of limit. The concept of limit is vital to differentiation. If a continuous function is differentiable, it has a limit at every point of its domain. So we can relate a variable term with the limiting value of the function using the above definition. Why the differentiation is the limit of some quantity is a meaningless question. We have defined it to be so as we have no other definition other than that. The same reasoning applies to the method of integration. We define the integration as the limit of a sum. This is the way calculus works and it gives the most precise results. We know that the area of a circle is 2(pi)r(squared). But why do we accept it? we accept it because calculus has proved it to be so. We can use calclus to determine curvature of space-time, volume of a sphere, rate of change of any continuously changing variable.
Fundamental theorem of calculus
The fundamental theorem of calculus relates differential of a function to its integral. The fundamental theorem says differentiation is the opposite operation of integration.
We can bring back original function by differentiating the same integrated fuction.
Mean Value theorem
The secant and tangent line are parallel to each other which have the slope
defined by the differentiation at point c. I have not given definition of the derivative of a function yet. The differentiation of a function is the rate of change at a given point which the independent variable takes. The rate of change is the slope of the curve at that particular point . The slope is determined by some limiting value. To be more precise, it is limit of a fraction. We take every small quantities and continuously devide them. In the limit the quotient of the division defines the slope of the curve at particular point.A formal definition can be given :
This is equation that has changed the world. Differentiation is the essence of calculus.
Riemann integration , in general, is defined as a limit of sum. That is, we divide the area under a curve into thin rectangles of small width. Then we let the width go to zero and sum all the infintesimall rectangles. In the limit the number of rectangles will be infinite and we will get the total area under the curve f(x). Integration can be defined in a number of ways, each of which is equivalent. Lebesque integration is another such method. Lebesue integration is more powerful method than Riemann integration as the latter can not be applied everywhere. The explanation of Lebesque integration somewhat lengthy, which at the moment best left. A satement can clarify the difference between Riemann and Lebesque integration:
" Suppose you go to a restaurant and you have got some bills to pay. You take the bills and put the coins out of your pockets in order you find them untill you reach total sum. That is Riemann Integration. You can pay the bills in another way: you take the bills and colllect coins into little heaps and then pay the total sum by giving all the individual heaps. This is Lebesque integration "
The two end points are the lower and upper limit of integration. The number of rectangles is ratio of the difference of upper limit and the lower limit to the width of each rectangle. Integration is thus an infinte sum or limit of a sum of small quantities with the proviso(condition) that the limit exists.
Some derivatives of functions
Different function have different characterstics in terms of how the function changes value. Each function has its own derivative. These derivatives can be found by using difinition of differentiation. These derivatives are to be memorized if you want to be good at calculus. The differentiability of a function depends on the continuity of a function. If a function is continuous in some interval [a,b], it is usually differntiable at that interval where it is continuous. In other word limit should exist at the point where we want to find the slope of the curve defined by the function.
Why do we need calculus anyway? Calculus help us calculate rates of change like velocity or acceleration. Integral calculus help us find area under a curve. All the higher mathematics is founded upon principles of calculus. It is such an important tool that we can not dispense with it. Its applications in physics and engineering are pervasive.
Taylor's theoremIt is perhaps the most useful theorem both in mathematics and physics. Taylor's theorem is used to represent a function in terms of its derivatives. We can always approximate a function in some neighborhood of a point by taylor's theorem. The condition is that the function must be analytic at the point. "To be analytic" means the function is differentiable at the point.
Using taylor's theorem we can linearize a curve at a point. The behaviour of the curve will be linear as the distance is small. This is because we can ignore higher order terms in the Taylor's expansion.
Integration by partsIf we have combination of two functions as the integrand we can use integration by parts method to evaluate such integral. The general formula for this is given by :
Let us now consider an example below:
Differential equation relates a function with its derivative. So the solution to the equation is a function. Differential equations can contain an independent variable and one or more
derivatives of the dependent variable. Differential equation can also contain partial differential of a function of several variables. In that case the equation is known as partial differential
equation. First we focus on ordinary differential equation.
Ordinary differential equation can be classified into linear and nonlinear differential equation. Each of these can be homogeneous and non-homogeneous. If there is a function of independent variable separately in ordinary differential equation , then it is non -homogeneous differential equation. Otherwise it is homogeneous ordinary differential equation. The highest derivative of dependent function is the order of the equation. And the power to the highest order differential is known as the degree of the equation. Linear differential has special form which makes it linear in the dependent variable. This means linear combination of two or more solutions of linear differential equation is also a solution of the same equation.
The first equation as shown in the figure is non homogeneous second order equation. It contains a function of independent variable x on the RHS. The second equation is homogeneous
equation of third order because the there is no function of x or f(x) = 0. Third equation is non-linear non-homogeneous fourth order differential equation. There are infinite variety of
such equations. The most useful differential equation encountered in electrical engineering is second order non-homogeneous linear differential equation. There is no specific method
to solve all kinds of differential equation ( ODE, PDE). A specific solution corresponds to a class of differential equations of the same form. Here
are some methods and solutions of ordinary differential equation. That is why differential equations are of great interest to mathematicians and engineers. Non-linear differential equations
are harder to solve than linear equations. Einstein Field Equation is non linear differential equation but it is partial differential equation. Schrodinger equation is linear partial differential equation.
Newton's second law of motion is second order linear differential equation, which involves second derivative of position with respect to time. The role of differential equation in physics and
mathematics is unimaginable. Without calculus the differential equation is hard to grasp , at least, intuit-ably. I have already written about basic idea like limit, infinitesimal in my thesis given in
other web page on waves and fields. Let us try to find a solution of a differential equation.
The method for solving the first order differential equation is to find characteristics equation assuming the solution to be of particular form(C.exp(at)). Then one constant in solution can be determined. The other constant C can be determined by initial condition that should be given or assumed also. This solution is called the general solution of the differential equation. Similarly higher order differential equation can be solved. General solution is composed of all solutions of particular form and are always satisfied by the differential equation. The above equation is a homogeneous equation and in this case the complementary solution is the same as general solution. If there is some function of x in the right hand side then we have to find the particular solution. The total solution namely general solution is the algebraic sum of both complementary and particular solution. Particular solution is a single solution.
Partial differential equation
There are various categories of partial differentials. They can be linear and non linear like ordinary differential equation. The unknown variable is usually function of space and time. As a result the solution depends both on space and time. The general form of partial differential equation is given below:
It comprises of all form of partial differential equations. As seen in the figure the equation contains partial derivatives of the function of dependent variable x, y. There are so varieties of partial differential equations that its quite impossible to list all of them and methods for solving them. Wave equation is an example of partial differential equation. Heat equation is also partial differential equation. In solving partial differential equation, we need to account for the initial conditions as well as boundary conditions which are the values of dependent variables at the boundary of the region enclosed by the domain of dependent variables. The domain can be subset of R(n) or euclidean n dimensional space. Euclidean space can be infinite dimensional that extends the notion of usual three dimensional physical space. Mathematics gives no objection to such abstract infinite dimensional space although it may be counter-intuitive. String theory predicts space has ten dimensions together with one time dimension. What do we know about dimension? Dimension , in usual sense, means any of three spatial extents of Euclidean space. But this is too narrow definition from mathematical point of view. Dimension is, fundamentally, any development of order. Order is certain relationship between points. The three spatial dimensions of Euclidean space are nothing but specific ordering of real numbers. But in mathematics order can be developed in many ways. Elements of any set can be ordered with specific relation defined on it. Each ordering is as valid as other. All this can be more analysed in order theory. Let us now try to solve a partial differential equation.
The above differential equation involves Laplacian(∇) in spherical polar coordinate system. What we are trying to find is the function (f) , the divergence of gradient of which in such coordinate system vanishes or the Laplacian vanishes. The function is dependent on three coordinate variable. The way to solve this equation is to separate variables into two functions f1 and f2. We assume that this decomposition works and carry out solving the equation. After inserting the decomposed function into the equation we can get , by algebraic manipulation, two separate equations. These two separate equations adds up to give zero as required. We choose an arbitrary constant ( l) to turn each of the equations into standard form. We get two solutions of f1 as stated. Solution is verified right when we substitute it in the equation of f1. The second equation can be decomposed into two equations each of which is related to different decomposed functions.
For angular part, we can further decompose the function f2 = Y into two functions and assign an arbitrary separation constant (m). This function f2 is seen to solve the original equation.
The P(lm) is associated Legendre polynomial. Y(lm) is the spherical harmonics which is a solution of Laplacian in spherical polar coordinates. This can be thought as spherical version of rectangular polar system. In rectangular polar we only have one angular variable or coordinates whereas in spherical polar we have two angular part. The method by which we have solved the equation is called separation of variables. Not every equation can be solved using separation of variables. Those functions which can be factored into multiple components of separate variables is only applicable for this method.
One of the most influential equations is the Black–Scholes equation. Black–Scholes equation is a second order partial differential equation which describes the dynamics of financial market in terms of assets (money, stock). The equation is given below:
The main factor or function in the equation is the price of the options V which is function of asset S and time. The solution for V can give riskless condition of financial market. The equation contains other terms and constant which should also be considered. The equation can be useful for businessman and investors when they enter real market. I don not have much economics related knowledge. I just mentioned the equation as it is also a partial differential equation and it has a form similar to diffusion equation. The method to transform the Black-Shole's equation to heat equation is given below.
Certain initial and boundary conditions are assumed. In the first step a slight altered form is created of the original equation. If the second term in the altered equation is manipulated by product rule of calculus , the same first equation can be attained. Then the change of variables are made and substituted in the equation. Following step 2 and 3 we arrive at the heat equation which has the special form as specified. Now we can try to solve the heat equation. The method is the separation of variables, which was explained earlier. The complete solution will be multiplication of two separate functions. It is the solution that led Fourier to discover Fourier series. So we can guess it has to have some relation to periodic functions.
One component of the solution (T) is found by separation of variables. It is an an exponential solution. Other solution is the sinusoidal functions, which is the second order equation of X of specific form. So the total solution is the product of these two solution. It has one periodic component. Complete exploration and method of solving the heat equation can be found here.
Notes and additional comments
Lebesque integration is defined in terms of supremum and infimum of sets. Supremum is the least upper bound whereas infimum is greatest lower bound. Every subset of a well
ordered set has a least upper bound. Upper bound of a subset S of a well ordered set (k, <=) is the element of K which is greater than or equal to every element of S. Least
upper bound is the smallest of all the upper bounds of S. If c is the least upper bound then no b > c is also an upper bound of S. If any b is also an upper bound then c is not the least upper bound.
We give completely symmetric definitions of "lower bound" and "least element". If A is a set of numbers and b is a number, we say that b is an lower bound for A iff for every number x ε A we have x => b. A set which has a lower bound is said to be bounded below. An lower bound for A which belongs to A is a least element of A (or a minimum element). Notice that A can have only one least element: suppose b and c are least elements of A. It follows that b ε A, c ε A, and for any x ε A, x=> b and x => c. From this it follows that b => c and c => b, so b = c.
Since we have defined least element of a set, we can define the least upper bound of a set A as the least element of the set of upper bounds of A. Since any set has at most one least element, this shows that a set A has just one least upper bound if it has any at all. The least upper bound of A is also called the supremum of A. It can be written sup(A) or lub(A). Sets with no upper bound have no least upper bound, of course. The set of all numbers is an example. The empty set has no least upper bound, because every number is an upper bound for the empty set. We give symmetric definitions of greatest lower bound.