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Ch12_4

# Ch12_4 - 12.4 12.4 Cylindrical Spherical Coordinates...

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12.4 Cylindrical & Spherical Coordinates Contemporary Calculus 1 12.4 CYLINDRICAL & SPHERICAL COORDINATE SYSTEMS IN 3D Most of our work in two dimensions used the rectangular coordinate system, but we also examined the polar coordinate system (Fig. 1), and for some uses the polar coordinate system was more effective and efficient. A similar situation occurs in three dimensions. Mostly we use the 3–dimensional xyz–coordinate system, but there are two alternate systems, called cylindrical coordinates and spherical coordinates, that are sometimes better. In two dimensions, the rectangular and the polar coordinate systems each located a point by means of two numbers, but each system used those two numbers in different ways. In three dimensions, each of the coordinate systems locates a point using three numbers, and each system uses those three numbers in different ways. Fig. 2 illustrates how three numbers are used to locate the point P in each of the different systems, and the rest of this section examines the cylindrical and spherical coordinate systems. x y x y P(x,y) Rectangular ! x y P(r, ! ) Polar Fig. 1 Fig. 2 x y z P( r, ! , z ) z ! r Cylindrical z ! r x y z P( " , ! , # ) # ! " Spherical x y z P( x, y, z ) x y z Rectangular x y z # ! "

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12.4 Cylindrical & Spherical Coordinates Contemporary Calculus 2 CYLINDRICAL COORDINATES Cylindrical coordinates are basically "polar coordinates with altitude." The cylindrical coordinates (r, θ , z) specify the point P that is z units above the point on the xy–plane whose polar coordinates are r and θ (Fig. 3). Example 1 : Plot the points given by the cylindrical coordinates A(3, π /3, 1), B(1, 0, –2), and C(2, 180 o , –1). Solution: The points are plotted in Fig. 4. Practice 1 : On Fig. 4, plot the points given by the cylindrical coordinates P(3, π /6, –1), Q(3, π /2, 2)), and R(0, π , 3). We can start to develop an understanding of the effect of each variable in the ordered triple by holding two of the variables fixed and letting the other one vary. Fig. 5 shows the results in the rectangular coordinate system of fixing x and y (x = 1, y = 2) and letting z vary: we get a vertical line parallel to the z–axis. Similarly, in the rectangular coordiante system, when we fix x and z (x = 1, z = 3) and let y vary we get a line parallel to the y–axis. We can try the same process in the cylindrical coordinate system. Fixing two of the variables and letting the other one vary: In the cylindrical coordinate system, if we fix r and θ (r = 2, θ = π /3 = 60 o ) and let z vary (Fig. 6a) we get a line parallel to the z–axis. If we fix r and z (r = 2, z = 3) and let θ vary (Fig. 6b) we get a circle of radius 1 centered around the z–axis at a height of 3 units above the xy plane. Fig. 3 x y z P( r, ! , z ) z ! r 3D Cylindrical 2D Polar ! x y P( r, ! ) r x y z A B C Fig. 4 Fig. 5: Rectangular coordinates: two variables fixed line (1, 2, z) x y z 1 2 x = 1 y = 2 z varies line (1, y, 3) 1 3 x y z x = 1 z = 3 y varies line (x, 2, 3) y = 2 z = 3 x varies 1 3 x y z
12.4 Cylindrical & Spherical Coordinates Contemporary Calculus 3 It we fix θ and z ( = π /3 = 60 o , z = 3) and let r vary (Fig. 6c) we get a line that is always 3 units above the xy plane.

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