 What we're going to do now is we're going to start the derivation for a stationary fluid, and that is the basic equation for fluid statics. What we'll do is we'll begin with our coordinate system and we're going to consider a chunk of fluid and we'll be looking at the pressure forces on that chunk of fluid. So we have a chunk of fluid size dx, dy, dz and we're going to assume that the pressure at the center is given by little p at x, y, and z. Now we're going to consider to begin with only the y component of pressure forces but on the back surface there would be a pressure force and it would act like that and on the front surface there would be another pressure force and remember pressure we said acts normal to the surface and so it would be acting inward into the chunk of fluid. So what I'm going to do I'm going to use a Taylor series expansion in order to express or expand the pressure that would be on both of those two faces, the left face and the right hand face. Okay so there we have the pressures acting on the left hand surface and the right hand surface and in case you're wondering this came through a Taylor series expansion and so I'll just do a little bit of a math aside now to explain where I was able to get those expansions or those terms for pressure. Okay so there what we've done is we've used the Taylor series expansion and so here's the Taylor series expansion and we've done this for the left hand surface and the right hand surface and so that's pl and pr and what you can see is that we only go to the first derivative and then we have these would be the higher order terms here which would be second and third order and we're neglecting those and with that we come up with an expression for the pressure on both the left hand face and the right hand face that if we look back you can see we have that within our expression for the pressure and in case you're wondering where the pressure came from that is where it came from. So with that what we can do is we can continue in that method and we've done that here for the left and the right in the y direction now what we'll do we continue on with the x and with the z direction and what we can do is rewrite or expand it for all three faces. So what we're going to do is we're going to write out the vector surface force due to pressure and we'll do this for the x direction the y and the z and for this we're multiplying by the area that the x direction pressure is acting on which is dy dz and then on the face on the upper side in the x direction let's go back to our cube of fluid which was right here so we're now dealing with the x direction we said x was positive in this way so the back face is back here the front face is here but notice that in this case we have the unit vector is i in that direction and here we're doing a dot product but the unit vector here is in the minus i direction and consequently we need to carry that through when we do our expansion and so we have minus i for the unit vector and we'll do that for the other surfaces as well. Okay so we get that for the surface the vector surface force due to pressure and what we'll notice to begin with you'll notice here this pressure and this pressure they cancel out because we have the minus i with the unit vector and similarly this and this and this and this so those pressures disappear and the other thing is we have dx over 2 and dx over 2 multiplied by dy dz dy dz well those are the same so they add up and then instead of dx over 2 you just get dx and so what we can do is we can rearrange this and we get the vector surface force due to pressure as being the following and it's multiplied by dx dy dz which happens to be the volume of the fluid element and the other thing that we can notice here if i pull the minus sign out and rewrite this this term here is nothing more than the gradient of pressure and remember the gradient operator takes a scalar and converts it into a vector field so what we can do is we can rewrite this so there's the gradient operator and what we can do is we can rewrite our surface forces due to pressure in that manner and then what i'm going to do i'm going to divide both sides by dv and i'm going to move the minus sign over so i'm going to have the gradient of p on the left and that will be equal to dfs by dv and i'm going to define this as a new force i'll do it as a little f and just like in thermodynamics when you have intensive properties that will be a little f which will denote force per unit volume so that is the equation the beginning of the equation for fluid statics we have to look at a couple of other things and that will be body forces as well as any shear that might be within the fluid but there are a couple of comments that i want to make here so what we can say is that the pressure gradient is the negative of surface force per unit volume that's what this equation here is telling us the other thing is that the pressure magnitude itself is not what is important what is really important is the change in pressure with distance and that is what the gradient operator tells us and it kind of makes sense if you have a big change in pressure with distance that's going to move the fluid and the fluid would not stay stationary but anyways it's it's the pressure changes with distance which is important and that is the first part of the fluid static equation we'll now continue on in the next segment looking at body forces as well as shear in the event that there is velocity fields within the fluid