in coordinate systems where the kinetic energy depends on the position and velocity of some generalized coordinates, q(t) and q˙(t), expressions for inertial forces become more complicated. For something as simple as an annulus... A smarter idea is to use a coordinate system that is better suited to the problem. TRIPLE INTEGRALS IN SPHERICAL & CYLINDRICAL COORDINATES Triple Integrals in every Coordinate System feature a unique infinitesimal volume element.

Accordingly, its volume is the product of its three sides, namely dV dx dy= ⋅ ⋅dz.

(A.6-13) vanish, again due to the symmetry. Instead

Orthogonal Curvilinear Coordinates 569 . ated by converting its components (but not the unit dyads) to spherical coordinates, and integrating each over the two spherical angles (see Section A.7).

For example in Lecture 15 we met spherical polar and cylindrical polar coordinates.

Complicated, isn’t it!

In this section we will define the spherical coordinate system, yet another alternate coordinate system for the three dimensional coordinate system.

In Rectangular Coordinates, the volume element, " dV " is a parallelopiped with sides: " dx ", " dy ", and " dz ".

In spherical polar coordinates we describe a point (x;y;z) by giving the distance r from the origin, the angle anticlockwise from the xz plane, and the angle ˚from the z-axis. Another important coordinate system is the spherical coordinate system, which is familiar be-cause we live on an approximately spherical object - the Earth (Figure 2.2b).

Lecture 23: Curvilinear Coordinates (RHB 8.10) It is often convenient to work with variables other than the Cartesian coordinates x i ( = x, y, z). The first goal, then, is to relate the work of inertial forces (P imr¨ δr) to the kinetic energy in terms of a set of generalized coordinates. The off-diagonal terms in Eq. TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES 2 Z 1 2 Z p 4 x2 2 p 4 2x dydx+ Z 1 1 Z p 1 x2 p 4 x dydx+ [There are two more integrals to write down.] A.7 ORTHOGONAL CURVILINEAR COORDINATES

where the dependence of the unit vector ˆr on the parameters θ and φ has been made explicit.
This coordinate system is described by the three parameters {r,#,'}, the radius, the polar angle, and the az-imuthal angle, respectively (Figure 2.2a).

The coordinate \(θ\) in the spherical coordinate system is the same as in the cylindrical coordinate system, so surfaces of the form \(θ=c\) are half-planes, as … It can be very useful to express the unit vectors in these various coordinate systems in terms of their components in a Cartesian coordinate system.

This coordinates system is very useful for dealing with spherical objects.
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