Page 375 - Advanced engineering mathematics

11.3 Vector Fields and Streamlines 355

Given a vector field F in 3-space, a streamline of F is a curve with the property that, at
each point (x, y, z) of the curve, F(x, y, z) is a tangent vector to the curve.
If F is the velocity field for a fluid flowing through some region, then the streamlines
are called flow lines of the fluid and describe trajectories of imaginary particles moving
with the fluid. If F is a magnetic field the streamlines are called lines of force. Iron filings
put on a piece of cardboard held over a magnet will align themselves on the lines of force.

Given a vector field, we would like to find all of the streamlines. This is the problem of
constructing a curve through each point of a region of space, given the tangent to the curve at
each point. To solve this problem suppose that C is a streamline of F = f i + gj + hk.Let C have
parametric equations x = x(ξ), y = y(ξ), z = z(ξ). A position vector for C is
R(ξ) = x(ξ)i + y(ξ)j + z(ξ)k.
Now

R (ξ) = x (ξ)i + y (ξ)j + z (ξ)k

is tangent to C at (x(ξ), y(ξ), z(ξ)) and is therefore parallel to the tangent vector F(x(ξ),
y(ξ), z(ξ)) at this point. These vectors must therefore be scalar multiples of each other, say
R (ξ) = tF(x(ξ), y(ξ), z(ξ)).

Then
dx dy dz
i + j + k =
dξ dξ dξ
tf (x(ξ), y(ξ), z(ξ))i + tg(x(ξ), y(ξ), z(ξ))j + th(x(ξ), y(ξ), z(ξ))k.
Equating respective components in this equation gives us
dx dy dz
= tf, = tg, = th.
dξ dξ dξ
This is a system of differential equations for the parametric equations of the streamlines. If f , g
and h are nonzero this system can be written as
dx dy dz
= = .
f g h

EXAMPLE 11.5
2
We fill find the streamlines of F(x, y, z) = x i + 2yj − k.If x and y are not zero, the streamlines
satisfy
dx dy dz
= = .
x 2 2y −1
These differential equations can be solved in pairs. First integrate
dx
=−dz
x 2
to get
1
− =−z + c
x

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October 15, 2010 16:11 THM/NEIL Page-355 27410_11_ch11_p343-366