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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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|dw:1440220440808:dw| This is a volume element of in an imcompressible fluid flowing at a distance x,y,z from the respective axes In the book it is given coordinates of P as x,y,z but shouldn't they be (x+dx),(y+dy),(z+dz) or (dx),(dy),(dz) if we consider the element at origin Let \[\vec V=V_{x}\hat i+V_{y}\hat j+V_{z}\hat k\] be the velocity of fluid at a point P(x,y,z) Mass of fluid flowing through face ABCD in unit time is \[=V_{x}(dy.dz)\] Now the problem is right in starting, this equation is dimensionally incorrect, the left side is having a unit of mass but right side is jut made up of velocity and length elements
Maybe what they mean by P(x,y,z) is \[\vec V \equiv \vec V(x,y,z)\] A vector point function \[\vec V=V_{x}(x,y,z)\hat i+V_{y}(x,y,z)\hat j+V_{z}(x,y,z)\hat k\] But using a notation like this makes u think the coordinates of P are x,y,z but that's impossible if the length is dx,dy,dz how can it have coordinates x,y,z It must have either (dx,dy,0) or (x+dx,y+dy,0) Depending on where we taking our volume element (point P lies in the x-y plane, so z=0)

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In the Gospel of Mary, V is defined as \(\vec V = \rho \vec v\) with units \(\frac{kg}{m^2 \ s}\) and so flow rate through "unit area" of surface with normal \(\hat n\) is \(\vec V \bullet \hat n\) which has units \(\frac{kg}{m^2 \ s}.\frac{m^2}{1}=\frac{kg}{s}\) http://www.amazon.co.uk/Mathematical-Methods-Physical-Sciences-Mary/dp/0471365807 it's only 3-4 pages so i'll copy them if you are using a different text. [and as it will come with a recommendation to buy, i reckon the copyright lawyers will forgive me.!]
and how does \[V_{x}((x+dx),y,z)=V_{x}+\frac{\partial V_{x}}{\partial x}.dx\]
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thanks, that was a lot helpful

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