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.
To determine the scale factors for an orthogonal coordinate system, we need to calculate the partial derivatives of the coordinate transformation equations. The scale factors are given by the magnitude of these partial derivatives.
Let's start by calculating the partial derivatives for each coordinate transformation equation:
Partial derivative of x (with respect to s):
∂x/∂s = 2t
Partial derivative of x (with respect to t):
∂x/∂t = 2s
Partial derivative of y (with respect to s):
∂y/∂s = 2s
Partial derivative of y (with respect to t):
∂y/∂t = -2t
Partial derivative of z (with respect to v):
∂z/∂v = 1
Now, we can calculate the magnitude of these partial derivatives to obtain the scale factors:
Scale factor in the s-direction:
√( (∂x/∂s)^2 + (∂y/∂s)^2 + (∂z/∂s)^2 ) =
√( (2t)^2 + (2s)^2 + 0^2 ) =
√(4t^2 + 4s^2) =
2 √(t^2 + s^2)
Scale factor in the t-direction:
√( (∂x/∂t)^2 + (∂y/∂t)^2 + (∂z/∂t)^2 ) =
√( (2s)^2 + (-2t)^2 + 0^2 ) =
√(4s^2 + 4t^2) =
2 √(s^2 + t^2)
Scale factor in the v-direction:
√( (∂x/∂v)^2 + (∂y/∂v)^2 + (∂z/∂v)^2 ) =
√( 0^2 + 0^2 + 1^2 ) =
1
Therefore, the scale factors for the coordinate system (s,t,v) are:
- Scale factor in the s-direction: 2 √(t^2 + s^2)
- Scale factor in the t-direction: 2 √(s^2 + t^2)
- Scale factor in the v-direction: 1