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If x = 0 and y > 0, then q = p /2 If x < 0 and y > 0, then q = p + Arctan ( y /x) If x < 0 and y = 0, then q = p If x < 0 and y < 0, then q = p + Arctan ( y /x) If x = 0 and y < 0, then q = 3p /2 If x > 0 and y < 0, then q = 2p + Arctan ( y /x)
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Consider a point P = (x,y,z) in Cartesian three-space How can we find the radius coordinate r of the point P in spherical coordinates How can we find the vertical angle coordinate f of the point P in spherical coordinates
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To find the spherical radius, we use the formula r = (x2 + y2 + z2)1/2 To find the spherical vertical angle, we use the formula f = Arccos [z / (x2 + y2 + z2)1/2] If we already know the radius r, then we have f = Arccos (z /r)
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PART
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CHAPTER
Relations in Two-Space
If you ve taken the course Algebra Know-It-All, you ve already had some basic training on relations and functions They ve been mentioned a few times in this book as well Let s look more closely at how relations and functions behave in two-space
What s a Two-Space Relation
A relation is a special way of assigning, or mapping, the elements of a source set to the elements of a destination set In two-space, both the source and destination sets usually consist of numbers The sets might be identical, partially overlapping, or entirely disjoint For example, we might have a relation between the set of negative integers and the set of positive integers, or the set of positive real numbers and the set of all real numbers, or the set of all real numbers and itself
Ordered pairs Any point in the Cartesian plane or the polar plane can be uniquely represented by an ordered pair in which a value of the independent variable (an element of the source set) is listed first, followed by a value of the dependent variable (an element of the destination set) The domain is the set of all values of the independent variable for which the relation produces defined values of the dependent variable The range is the set of all values of the dependent variable that come from the elements of the domain Here s an example of a relation written as a set of ordered pairs:
{(3,2),(4,3),(5,4),(6,5)} The domain of this particular relation (let s call it set D) is the set of first numbers in the ordered pairs Therefore D = {3,4,5,6}
Relations in Two-Space
The range (let s call it set R) of the relation is the set of second numbers in the ordered pairs, so R = {2,3,4,5}
Injection, surjection, and bijection Imagine a relation between numbers x in a set X and numbers y in a set Y Suppose that each number x in set X corresponds to one, but only one, number y in set Y Also suppose that no number in Y has more than one mate in X (There might be some numbers in Y without any mate in X) A relation of this type is called an injection In some older texts, it s called one-to-one Now imagine a relation that assigns the elements of set X to the elements of set Y so that every element of Y has at least one mate in X This type of relation is called a surjection Set Y is completely spoken for A surjection is sometimes called an onto relation, because it maps (assigns) the values from set X completely onto the entire set Y Finally, imagine a relation that is both an injection and a surjection This type of relation is called a bijection In older texts, you might see it referred to as a one-to-one correspondence (not to be confused with one-to-one, which means an injection) A bijection assigns every value of x in set X to a unique value of y in set Y Conversely, every y in set Y corresponds to a unique value of x in set X In this context, a unique value means one and only one value or exactly one value Example 1 Relations are commonly represented by equations Here s an example of a simple two-space relation that subtracts 1 from every value in the domain to generate values in the range:
y=x 1 This relation could describe a one-to-one correspondence between the elements of the domain X = {3,4,5,6} and the elements of the range Y = {2,3,4,5} which we saw a few moments ago If we allow the domain of the relation to extend over the entire set of real numbers, then the range also covers the entire set of real numbers When we put specific values of x into the equation, we get results such as the following: If x = 13, then (x,y) = ( 13, 14) If x = 16, then (x,y) = ( 16, 26) If x = 0, then (x,y) = (0, 1) If x = 1, then (x,y) = (1,0) If x = 3/2, then (x,y) = (3/2,1/2) If x = 81/2, then (x,y) = [81/2,(81/2 1)]