Question

vector fields

Answer 1

suppose a vector field \(\vec{F}\)
perhaps describing force or velocity acting on a object at a particular point in space...

its path would be a curve which is tangent to \(\vec{F}\) at every point...( i guess)

suppose \(\vec{F}\) =<−y, x>
we can easily find the path followed,
we just need an equation with slope \(\frac{x}{-y}\)

\[\frac{dy}{dx}=\frac{-x}{y}\\ydy=-xdx\\y^2/2=-x^2/2+c\\x^2+y^2=C\]

which is a circle...

Answer 2

how do i do the same thing in 3D?
\[\vec{F} = < X(x,y,z)~ , Y(x,y,z)~,Z(x,y,z)>\]

(i want to find a curve whose tangent vector at every point is paralell to \(\vec{F}\) )

Answer 3

i do not have a specific example in mind...

Answer 4

.