Use a parametrization to find the flux double integral from s ( f. n) dstd of f= 3zk across the portion of the sphere x^2+y^2+z^2=a^2 where y is positive in the direction away from the origin. * std = standard deviation symbol. the flux is

Following are the calculation to the given qestion:

Consider the vector function:  

Normal vector:

When the dot product of vectors= :

Using sphereical coordinates,

So, the dot product is

The area element on the surface of the sphere of radius:

Now, the surface integral:  

Therefore, the required flux is

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Parameterize the part of the sphere you're interested in, , by

with and . The domain of guarantees that we're dealing with only positive values of .

Then the flux of over is given by the surface integral

where the order of the partial derivatives is chosen to make sure we pick the normal vector to pointing in the positive direction.

We have

Then the integral reduces to

Substitute , , then

which leaves us with a flux of .