target="_blank" rel="nofollow" href="#fb3_img_img_696d4216-81b6-5bc3-9369-9e6644a22b2a.png" alt="equation"/>
where a change of variable to
(2.598)
all is left is taking the inverse hyperbolic cotangent of both sides, according to
(2.599)
where retrieval of the original (dummy) variable x unfolds
Equation (2.600) is depicted in Fig. 2.15 b; it is not defined within [−1,1] since x – 1 < 0 in that range would compromise existence of the logarithm. Outside said range, one notices that
(2.601)
based on Eq. (2.600) – so x = −1 drives the behavior of cotanh−1 x toward −∞, in the neighborhood of −1; by the same token,
(2.602)
so cotanh−1 x tends to (positive) infinite when x = 1 is approached – meaning that x = 1 serves as vertical asymptote as well. For the remainder of its domain, this inverse function is monotonically decreasing in either interval]−∞,−1[ or]1,∞[; when x →−∞, one obtains
(2.603)
stemming from Eq. (2.600) – and similarly when x → ∞, i.e.
(2.604)
because x + 1 ≈ x and x − 1 ≈ x, thus indicating that the horizontal axis plays the role of (single) horizontal asymptote when x grows unbounded.
3 Vector Operations
As indicated previously, a vector u is defined as a quantity possessing both a magnitude and a direction; the said magnitude is regularly denoted by ‖ u ‖, while information on the direction is often conveyed graphically – or else encompasses angles formed with the axes in some reference system. Two vectors, u and v, are said to be equal when their magnitudes are identical, i.e. ‖ u ‖ = ‖ v ‖, and also point in the same direction; however, they do not need to have the same origin.
A much more convenient way of handling vectors resorts, however, to their decomposition along the three directions of space in a typical Cartesian R3 domain, according to
and
here jx, jy, and jz denote unit, orthogonal vectors of a Cartesian system, defined as
(3.3)
(3.4)
and
(3.5)
– while
(3.6)
and
(3.7)
define u and v, respectively, via their coordinates.
According to Pythagoras’ theorem,
and likewise
this is a more general form than Eq. (2.431), yet it relies on application of the aforementioned theorem twice. In fact,
abides to Eq. (2.431), as long as ux and uy denote the projections of u onto the x‐ and y‐axis, respectively, and uxy denotes the projection of u onto the x0y plane; further application of Eq. (2.431) then supports
where uz denotes the projection of u onto the z‐axis. Insertion of Eq. (3.10) transforms Eq. (3.11) to
(3.12)
that
