Determine the scale factors for an orthogonal coordinate system (s, t, v) whose coordinates are related to the Cartesian by the 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 an orthogonal coordinate system, we need to calculate the partial derivatives of the new coordinates with respect to the Cartesian coordinates. In this case, we have three new coordinates (s, t, v) related to the Cartesian coordinates (x, y, z) as follows:
x = 2st,
y = s^2 - t^2,
z = v.
Let's calculate the scale factors for each coordinate:
For s:
To find the scale factor for s, we differentiate x with respect to s and keep the other variables constant.
∂x/∂s = 2t.
For t:
To find the scale factor for t, we differentiate x with respect to t and keep the other variables constant.
∂x/∂t = 2s.
For v:
To find the scale factor for v, we differentiate z with respect to v and keep the other variables constant.
∂z/∂v = 1.
The scale factors for this orthogonal coordinate system are:
Scale factor for s: 2t,
Scale factor for t: 2s,
Scale factor for v: 1.
Now, let's derive an expression for the square of the arc element, ds^2, corresponding to this coordinate system.
The square of the arc element in an orthogonal curvilinear coordinate system is given by the expression:
ds^2 = (∂x/∂s)^2 * ds^2 + (∂y/∂t)^2 * dt^2 + (∂z/∂v)^2 * dv^2.
Substituting the scale factors we calculated earlier:
ds^2 = (2t)^2 * ds^2 + (2s)^2 * dt^2 + (1)^2 * dv^2.
Simplifying:
ds^2 = 4t^2 * ds^2 + 4s^2 * dt^2 + dv^2.
This expression represents the square of the arc element for the given orthogonal coordinate system.