00:01
All right, so for this question, as basically a vector calculus question, involving flux integrals.
00:10
All right.
00:12
So let's start with what we're given.
00:14
We're given a velocity vector here for x, y, and z, which is 2x, i, minus 3, y, and j, plus z times k.
00:26
Right so this is our velocity vector on the fluid particle they say right at any point x y and z and what we need to do is we need to find the net volume of fluid that passes in the upward direction through a portion of the plane that's modeled by x plus y plus z equals one that's in the first octet in one second, right? so let's break that down.
01:02
So basically what we need to do is we need to, so we need to find the limits for the x and y, right? and then after that, we have to compute the flux integral to find the area in that first octant.
01:19
And then after we compute the flux integral, we have to take that area and multiply it times the volume for our flow rate.
01:27
Because flow rate, the flow rate, which is basically the net volume, so the flow rate volume is basically equal to the area of our whatever is underneath the plane, according to our velocity vector.
01:50
And we have to multiply that by our velocity vector.
01:52
And that's how we basically get our flow rate.
01:55
So that's all the steps you have to do for the first part.
01:58
So let's get started.
01:59
So the first thing we're given is x plus y plus z equals 1 and we can actually simplify that into z equals 1 minus x minus y to basically, what do you call? so that's easier for us to basically see, right? and then next what we have to do is find our limits.
02:29
So our x is going from basically from zero to.
02:33
To 1, right? so 0, x, 1, and our x, sorry, our y, right, our y is going to be going from 0 to y to 8 minus x, right? so now that we have these limits, and the way i got these is basically just, is just each individual component, right? so we look at z equals 1 minus x minus y.
03:16
And when we actually simplify that down, right, this is basically what parametization is.
03:25
So when we plug in zero for everything, then x has to be between 1, sorry, not 8, 1 minus x.
03:35
When we plug in 0 for y, then x has to be between, and for y and z, then x has to be between 0 and 1.
03:41
And when we plug in 0 for z for the y value, and we know x has to be at least 1, so we can't say from 0 to 0.
03:53
So everything in moving 0 and 1 for y as well.
03:56
So we have to represent that by saying 1 minus x, since we already have a balance for x.
04:03
Right...