For the vector field defined by ,
Using Gauss’ divergence theorem,
[Note that the details of the sphere are irrelevant - for this source-free vector field, the net flux through any simple closed surface will be zero!]
In the x-y plane a vector field is defined by .
Green’s theorem states that, in any simply connected
domain,
where C is a simple closed path that
encloses the region D.
But D is a unit square, whose area is
A = 1. Therefore
NO, F is not conservative. |
The velocity field of a fluid is
.
Find the flux Q of the fluid through
the hemisphere
in a direction outward from its centre.
[Hint: use a spherical polar coordinate grid
, with
But, everywhere on the hemisphere,
are all positive.
For an outward pointing normal vector (pointing away from the
origin), we must select the '+' sign.
Therefore the flux is
BONUS QUESTION
Find the potential function for such that everywhere along the y-axis.
But
everywhere along the y-axis
Therefore the required potential function is
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