Find spherical coordinates for the point. 12.7ġ7 The point is given in rectangular coordinates.Įxample 2 The point is given in rectangular coordinates. 12.7Įxample 1 SOLUTION From Equations 1, we have: 12.7ġ6 Thus, the point (2, p/4, p /3) is in rectangular coordinates.Įxample 1 SOLUTION Thus, the point (2, p/4, p /3) is in rectangular coordinates. 12.7Įxample 1 SOLUTION We plot the point in Figure 6. Plot the point and find its rectangular coordinates. ![]() ![]() 12.7ġ3 The point (2, p/4, p /3) is given in spherical coordinates.Įxample 1 The point (2, p/4, p /3) is given in spherical coordinates. COORDINATES Also, the distance formula shows that: r 2 = x2 + y2 + z2 We use this equation in converting from rectangular to spherical coordinates. COORDINATES So, to convert from spherical to rectangular coordinates, we use the equations x = r sin f cos y = r sin f sin z = r cos f 12.7ġ2 Also, the distance formula shows that: r 2 = x2 + y2 + z2 12.7įrom triangles OPQ and OPP’, we have: z = r cos f r = r sin f However, x = r cos y = r sin 12.7ġ1 SPH. The relationship between rectangular and spherical coordinates can be seen from Figure 5. 12.7ħ The graph of the equation = c is a vertical half-plane.Ĩ The equation f = c represents a half-cone with the z-axis as its axis. This is the reason for the name “spherical” coordinates. 12.7Ħ SPHERE For example, the sphere with center the origin and radius c has the simple equation r = c. The spherical coordinate system is especially useful in problems where there is symmetry about a point, and the origin is placed at this point. ![]() f is the angle between the positive z-axis and the line segment OP. is the same angle as in cylindrical coordinates. r = |OP| is the distance from the origin to P. The spherical coordinates (r, , f) of a point P in space are shown in Figure 1. It simplifies the evaluation of triple integrals over regions bounded by spheres or cones. SECTION 12.7 TRIPLE INTEGRALS IN SPHERICAL COORDINATESĪnother useful coordinate system in three dimensions is the spherical coordinate system. 1 TRIPLE INTEGRALS IN SPHERICAL COORDINATES
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