Page 86 - Intro to Tensor Calculus

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The above definition of a dot product in V N can be used to define unit vectors in V N .

Definition: (Unit vector) Whenever the magnitude of a vec-
i
tor A is unity, the vector is called a unit vector. In this case we
have
j
i
g ij A A =1. (1.3.28)

EXAMPLE 1.3-8. (Unit vectors)
j
i
2
In V N the element of arc length squared is expressed ds = g ij dx dx which can be expressed in the
i
dx dx j dx i
form 1 = g ij . This equation states that the vector ,i =1,... ,N is a unit vector. One application
ds ds ds
of this equation is to consider a particle moving along a curve in V N which is described by the parametric
i
i
i
equations x = x (t), for i =1,... ,N. The vector V = dx i ,i =1,... ,N represents a velocity vector of the
dt
particle. By chain rule differentiation we have
i
dx i dx ds dx i
i
V = = = V , (1.3.29)
dt ds dt ds
where V = ds is the scalar speed of the particle and dx i is a unit tangent vector to the curve. The equation
dt ds
(1.3.29) shows that the velocity is directed along the tangent to the curve and has a magnitude V. That is
2
ds 2 i j
=(V ) = g ij V V .
dt

EXAMPLE 1.3-9. (Curvilinear coordinates)
Find an expression for the cosine of the angles between the coordinate curves associated with the
transformation equations

x = x(u, v, w), y = y(u, v, w), z = z(u, v, w).