Calculus
Created | Updated Feb 19, 2007
The History
Calculus was first invented by Isaac Newton1, who promptly told no one about it. He called it the "Theory of Fluctions" (a name which somehow did not persist), and used it to uncover the laws of gravitation and motion. Howerver, most credit for the developement of Calculus lies with the mathemeticians who shared their insights. People such as Leibniz, whose notation is still in use.
What is Calculus
Calculus is a system that allows us to describe the world quantitavely. With it, we are no longer dependant on friendly digital descriptions with easy-to-use straight edges. First it frees us to take really tough bits of calculations and sort-of ignore the sticky bits and get an almost decent estimate. Then we break it down into alot of smaller bad estimates and add them up. Then calculus lets us use a practically infinite2 number of tiny estimates to get a smashingly accurate sum - usually without ever really facing the sticky bits of the calculation. In the final run-through calculus does all of this for us with friendly notation and a shockingly finite number of mathematical steps.
Principles
Calculus begins with three not-so-basic mathematical concepts. Limits, derivatives, and integrals. These, respectively, get really close to a non-existant number, find slopes of equations, and find areas under curves. They have many other less-obvious applications and properties.
Limit Theory
There are a great number of mathematical expressions wich do not, in fact, express anything. The most famous is a number divided by zero. This, in itself is meaningless.
But not all such 'meaningless' numbers are useless. The most famous is the case of the Frog who jumps half of the remaining distance to a goal with every jump. Of course the Frog will never make it to the goal, but in time, he will get as close to the wall as you could care to have him. Tell me how close you want him, and I can tell you how many times he must hop. This becomes imensly useful when one realizes that one can measure the trip of the frog to within a thousandth of a millimeter, even though the trip itself will never be completed. More specifically this becomes useful when it is applied to issues of greater use than obsessed frogs.
Specific uses of limits include calculating the exact digits of the number Pi, and dealing with the concept of infinity3.
Differentiation
Differentiation, or finding the derivative of an equation, in it's simplest form gives the slope4 of a line at any point. The line need not be straight, it only needs to be defined mathematically and continuous at the points of interest.
The second derivativeThe second derivative is the name given to the derivative of a derviative of a function. It has a name because saying "the derivative of the derviative of the function" too often is bad for your mental health. Similarly, the third derivative is the derivative of the second derivative, etc.
Famous Derivatives
The derivative of a line gives the slope of that line.
The derivative of a slope gives the concavity of the line.
The derivative of a constant number is zero.
The derivative of motion (a function of distance per time unit), is the change in distance over time, or the speed, at each moment.
the second derivative of a function of distance per time unit is the acceleration (the change in speed) at each moment.
the third derivative of motion is 'jerk' (the change in acceleration, what you feel when there is a change in acceleration).
Integral Calculus
Integration is essentially the process of differentiation in reverse. Integration is often called Antidifferentiation. Integrals are generally found by answering the question 'What equation would I have to differentiate to get the equation in front of me?'. Unfortunately, because the derivative of any constant is zero, an integral can only come within a constant of the original equation. So taking the integral of a slope gives the original line, plus or minus an unkown number. This often leads to alot of messy mathematics to pin down what this lost number was anyways. The most famous application of the integral is finding the area enclosed by a curve. In advanced caclulus this concept is extended to finding the volume enclosed by a surface. All of the basic geometric area and volume equations of shapes and solids can be validated using integral calculus. In fact, they validate eachother. Secondly, integral calculus can solve an equation for all possible values of a variable with a single calculation. This makes problems that would normally require thousands of figures, require only a few steps. A good example is a problem that has a variable changing every second.
Fun with Calculus
Mathematics below calculus has a great many problems that it simply will not attempt. Many of these are startlingly easy when the proper calculus is applied.
These include: (but are certaintly not limited to)
Proving that the area of a circle is pi times the radius squared.
Determining how far a thrown object actually travelled along the path of the arc in three dimensions.
Finding the centroid of an irregular region.
Finding the area or volume of an irregular object.
Determining the rate of change in volume of a solid object at any give moment while its various dimensions are growing a shrinking at different rates.
Gradients (more on slopes)
A gradient is the core of a three dimensional slope at a given point.
The gradient is a vector whose x, y and z components are the partial derivative of the function with respect to x, y and z, respectively. The gradient of f = df/dx * i + df/dy * j + df/dz * k = <df/dz,df/dy,df/dz>.
The Gradient of a surface is a vector pointing in the direction of the maximum increase. Multiply 5 your gradient by any other direction vector and the result is the directional derivative or simply the slope of the surface in your chosen direction.
The gradient is helpful when finding:
