Advanced Calculus fi..

Chapter 3 Vector Differential Calculus 189The curvilinear coordinate system defined by (3.42) and (3.43) is said to beorthogonal if the tangent vectors ar/au, ar/av, ar/aw at each point of D form atriple of mutually perpendicular vectors. It will be seen that when this is the case,important simplifications in the formulas are possible. The most commonly usedcurvilinear coordinates are orthogonal, and for this reason we shall confine attentionto this case. In this and the following section the coordinates will be assumed to beorthogonal.As a first consequence of orthogonality, we remark that (lla) arlau, (118) ar/av,(lly) ar/aw are a positive triple of mutually perpendicular unit vectors. HenceAccordingly, by (3.44,A similar reasoning applies to V G and V H ; we conclude that a V F, BVG, y V Hare also mutually perpendicular unit vectors andThe surfaces F = const, G = const, H = const must hence meet at right angles;they form what is called a triply orthogonal family of surfaces. Conversely, when thevectors V F, V G, V H are mutually perpendicular throughout D, the coordinatesmust be orthogonal (Problem 4 following Section 3.8).A curve in D can be described by equations: x = x(t), y = y(t), z = z(t), or,by (3.43), in terms of the curvilinear coordinates by the equations u = u:t), v =v(t), w = w(t). The element of arc ds on such a curve is defined by the equationHenceSince the coordinates are orthogonal, we conclude: