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Homework Statement
Trying to get my head around the physical interpretation of pathlines and the math that describes them. The physical explanation is simple enough, they are just the path a particle of fluid will follow through the vector field.
Homework Equations
In the vector field v=(u,v,w) at time t=0 a particle has position a=(a,b,c).
Writing its position as X(t)=(X(t),Y(t),Z(t)) the equations of motion are,
dX/dt = u(X,Y,Z,t)
dY/dt = v(X,Y,Z,t)
dZ/dt = w(X,Y,Z,t)
So I guess this just means we can get the velocity from the position or vice-versa.
This can be solved at least locally in time, and the solution would
be represented as X(t) = X(a, t) = (X(t), Y (t), Z(t)), where X(t) = X(a,b,c,t),
Y (t) = Y (a,b,c,t), Z(t) = Z(a, b, c, t).
This part has me confused, I think its saying solve the equations at t=0 when the position is X=a ?
The Attempt at a Solution
Trying this,
∫dX = ∫u(X,Y,Z,t)dt
X(t) = X(X,Y,Z,t) so this gives the X position
X(0) = X(a,b,c,0) + [STRIKE]f(y,z)[/STRIKE]
and similarly
Y(0) = Y(a,b,c,0) + [STRIKE]f(x,z)[/STRIKE]
Z(0) = Z(a,b,c,0) +[STRIKE]f(y,z)[/STRIKE] and the Y,Z positions