The spherical coordinates of a point P in 3-space are ρ (rho), θ, and ϕ (phi), where ρ is the distance from P to the origin, θ is the angle that t...The spherical coordinates of a point P in 3-space are ρ (rho), θ, and ϕ (phi), where ρ is the distance from P to the origin, θ is the angle that the projection of P onto the xy-plane makes with the positive x-axis, and ϕ is the angle between the positive z axis and the vector from the origin to P. When P has Cartesian coordinates (x,y,z), the spherical coordinates are given by
Given the difficulty of evaluating multiple integrals, the reader may be wondering if it is possible to simplify those integrals using a suitable substitution for the variables. The answer is yes, tho...Given the difficulty of evaluating multiple integrals, the reader may be wondering if it is possible to simplify those integrals using a suitable substitution for the variables. The answer is yes, though it is a bit more complicated than the substitution method which you learned in single-variable calculus.
Many problems possess natural symmetries. We can make our work easier by using coordinate systems, like polar coordinates, that are tailored to those symmetries. We will look at two more such coordina...Many problems possess natural symmetries. We can make our work easier by using coordinate systems, like polar coordinates, that are tailored to those symmetries. We will look at two more such coordinate systems — cylindrical and spherical coordinates.
Sometimes, you may end up having to calculate the volume of shapes that have cylindrical, conical, or spherical shapes and rather than evaluating such triple integrals in Cartesian coordinates, you ca...Sometimes, you may end up having to calculate the volume of shapes that have cylindrical, conical, or spherical shapes and rather than evaluating such triple integrals in Cartesian coordinates, you can simplify the integrals by transforming the coordinates to cylindrical or spherical coordinates. For this topic, we will learn how to do such transformations then evaluate the triple integrals.