Determine the scale factors for an orthogonal coordinate system ( s, t, v ) whose coordinates are related to the cartesion by following equations:
x = 2st:, y = s^2-t^2:, z = v
Also write down an expression for the square of the arc element.
To determine the scale factors for the orthogonal coordinate system (s, t, v), we need to find the partial derivatives of each Cartesian coordinate (x, y, z) with respect to each coordinate (s, t, v).
Given the equations relating the Cartesian coordinates (x, y, z) to the orthogonal coordinates (s, t, v) as follows:
x = 2st
y = s^2 - t^2
z = v
Let's calculate the partial derivatives:
1. For the x-coordinate:
∂x/∂s = 2t
∂x/∂t = 2s
∂x/∂v = 0
2. For the y-coordinate:
∂y/∂s = 2s
∂y/∂t = -2t
∂y/∂v = 0
3. For the z-coordinate:
∂z/∂s = 0
∂z/∂t = 0
∂z/∂v = 1
The scale factors for the orthogonal coordinate system are given by the magnitude of these partial derivatives:
h_s = √(∂x/∂s)^2 + (∂y/∂s)^2 + (∂z/∂s)^2 = √(2t)^2 + (2s)^2 + 0^2 = 2√(s^2 + t^2)
h_t = √(∂x/∂t)^2 + (∂y/∂t)^2 + (∂z/∂t)^2 = √(2s)^2 + (-2t)^2 + 0^2 = 2√(s^2 + t^2)
h_v = √(∂x/∂v)^2 + (∂y/∂v)^2 + (∂z/∂v)^2 = 0^2 + 0^2 + 1^2 = 1
Therefore, the scale factors for the orthogonal coordinate system are:
h_s = 2√(s^2 + t^2)
h_t = 2√(s^2 + t^2)
h_v = 1
To find the expression for the square of the arc element, we use the formula:
(ds)^2 = (h_s)^2*(ds)^2 + (h_t)^2*(dt)^2 + (h_v)^2*(dv)^2
Substituting the scale factors we found above:
(ds)^2 = (2√(s^2 + t^2))^2*(ds)^2 + (2√(s^2 + t^2))^2*(dt)^2 + (1)^2*(dv)^2
Simplifying:
(ds)^2 = 4(s^2 + t^2)*(ds)^2 + 4(s^2 + t^2)*(dt)^2 + (dv)^2