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Current time:0:00Total duration:4:04

Video transcript

so in the last video I talked about three-dimensional vector fields and I finish things off with this sort of identity function example where at an input input point XYZ the output vector is also XYZ and here I want to go through a slightly more intricate example so I'll go ahead and get rid of this vector field and in this example the X component of the output will be Y times Z the Y component of the output will be x times Z and the Z component of the output will be x times y so we'll just show this vector field and then we can start to get a feel for how the function that I just wrote relates to the vectors that you're seeing so you see some of the vectors are kind of pointing away from the origin some of them are pointing in towards the origin so how can we understand this vector field in terms of the function itself and a good step is to just zero in on one of the components so in this case I'll choose the Z component of the output which is made up of x times y and kind of start to understand what that should be the Z component will represent how much the vectors point up or down this is the z axis and the xy-plane here so I'll point the z axis straight in our face this is the x axis this is the y axis the values of x and y are going to completely determine that Z component so I'm going to go off to the side here and just draw myself a little XY plane for reference so this is my x value this is my Y value and I want to understand the meaning of the term x times y so when both x and y are positive their product is positive and when both of them are negative their product is also positive if X is negative and Y is positive the product is negative but if X is positive and Y is negative their product is also negative so with the should mean in terms of our vector field is that when we're in this first quadrant vectors tend to point up in the Z direction same over here in the third quadrant but over on the other two they should tend to point down so let's focus in on that first quadrant and try to look at what's going on so we see like this vector here applies this vector and all of them generally point upwards they have a positive z component so that seems in line with what we were predicting whereas over here which corresponds to the fourth quadrant of the XY plane the Z component of each vector tends to be down you know this and they're doing they're doing other things in terms of the X and y component is not just Z component action but right now we're just focusing on up and down and if you look over in the third quadrant they tend to be pointing up and that corresponds to the fact that x times y will be positive and you're looking at it all starts to align that way and because I chose a rather symmetric function you could imagine doing this where you you analyze also the the y component here you analyze the X component in terms of y&z and it's actually going to look very similar for understanding when the X component of a vector tends to be positive like up here or when it turns out to be negative like over here and same with when the y component of a vector tends to be negative or if the Y component tends to be positive and overall it's a very complicated image to look at but you can slowly piece by piece get a feel for it and just like with 2-dimensional vector fields a kind of neat thing to do is imagine that this represents a fluid flow so imagine like maybe air around you flowing in towards the origin here flowing out away from the origin there you know kind of be rotating around here and later on and multivariable calculus you'll learn about various ways that you can study just the function itself and just the variables and get a feel for how that fluid itself would behave even though it's a very complicated thing to think about it's even complicated to draw or use graphing software with but just with analytic tools you can get very powerful results and these kind of things come up in physics all the time because you're thinking in three-dimensional space and it doesn't just have to be fluid flow it could be a force field like an electric force field or gravitational force field where each vector tells you how a particle tends to get pushed and as we continue on with multivariable calculus you'll get to see lots more examples but hopefully this helps get a small feel for how you can go piece by piece and understand something that's kind of a complicated expression