 # Sphere Inside out Part - II

Shap Coordinates: Etymology Early (12th- and 13th-century) forms such as Hep and Yheppe point to an Old Norse rendering Hjáp of an Old English original Hēap = "heap", (of stones), perhaps referring to an ancient stone circle, cairn, or to the Shap Stone Avenue just to the west of the village.  Description The village has three pubs, a small supermarket, a fish and chip shop, an antique book shop, a butcher's shop, a primary school, a newsagent's, a coffee shop, a ceramic art studio called Edge Ceramics, a fire station, a bank (only open 4 hours a week), a shoe shop (New Balance factory shop) an Anglican church and 3 B&B/ Hostels. Major employers in the area are Hanson and Tata Steel. The civil parish of Shap (formerly Shap Urban Parish) includes the hamlet of Keld and parts of the granite works and limestone works, and has a population of 1,221. The parish shares a joint parish council with Shap Rural. Shap is on the route of the Coast to Coast Walk. Transport links

What Is A Differential Equation? A differential equation can look pretty intimidating, with lots of fancy math symbols. But the idea behind it is actually fairly simple: A differential equation states how a rate of change (a "differential") in one variable is related to other variables. For example, the single spring simulation has two variables: time t and the amount of stretch in the spring, x. If we set x = 0 to be the position of the block when the spring is unstretched, then x represents both the position of the block and the stretch in the spring. the rate of change in velocity is proportional to the position For instance, when the position is zero (ie. the spring is neither stretched nor compressed) then the velocity is not changing. On the other hand, when the position is large (ie. the string is very much stretched or compressed) then the rate of change of the velocity is large, because the spring is exerting a lot of force. What is a Solution to a Differential Equation? Initial Conditions

Geometry Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as Thales (6th Century BC). By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow. Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. In Euclid's time, there was no clear distinction between physical and geometrical space. Overview Practical geometry Axiomatic geometry Geometry lessons in the 20th century

Classifying Differential Equations When you study differential equations, it is kind of like botany. You learn to look at an equation and classify it into a certain group. The reason is that the techniques for solving differential equations are common to these various classification groups. And sometimes you can transform an equation of one type into an equivalent equation of another type, so that you can use easier solution techniques. First Order, Second Order, etc. The order of a differential equation is equal to the highest derivative in the equation. Linear vs. Linear just means that the variable in an equation appears only with a power of one. In math and physics, linear generally means "simple" and non-linear means "complicated". Recall that the equation for a line is y = m x + b where m, b are constants ( m is the slope, and b is the y-intercept). Homogeneous vs. This is another way of classifying differential equations.

Equation The first use of an equals sign, equivalent to 14x + 15 = 71 in modern notation. From The Whetstone of Witte by Robert Recorde (1557). The = symbol was invented by Robert Recorde (1510–1558), who considered that nothing could be more equal than parallel straight lines with the same length. Parameters and unknowns Equations often contain terms other than the unknowns. A system of equations is a set of simultaneous equations, usually in several unknowns, for which the common solutions are sought. has the unique solution x = −1, y = 1. Analogous illustration Illustration of a simple equation; x, y, z are real numbers, analogous to weights. A weighing scale, balance, or seesaw is often presented as an analogy to an equation. In the illustration, x, y and z are all different quantities (in this case real numbers) represented as circular weights, and each of x, y, and z has a different weight. Types of equations Identities , which is true for all values of θ. use the identity:

Mathematical Equations Solving Linear Equations Math series Linear Equation: a mathematical expression that has an equal sign and linear expressions Variable:a number that you don't know, often represented by "x" or "y" but any letter will do! Variable(s) in linear expressions Cannot have exponents (or powers) For example, x squared or x2 Cannot multiply or divide each other For example: "x" times "y" or xy; "x" divided by "y" or x/y Cannot be found under a root sign or square root sign (sqrt) For example: √x or the "square root x"; sqrt (x) Linear Expression: a mathematical statement that performs functions of addition, subtraction, multiplication, and division These are examples of linear expressions: These are not linear expressions: Solve these linear equations by clicking and dragging a number to the "other" side of the equal sign. More examples: Linear equation, solving example #1: Find x if: 2x + 4 = 10 Linear equation, solving example #2: Find x if: 3x - 4 = -10(using negatives) Linear equation, solving example #3: Math series

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