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Current time:0:00Total duration:11:05

Video transcript

what we will attempt to start to do in this video is take the surface integral take the surface integral of the function x squared over our surface where the surface in question the surface we're going to care about is going to be the unit sphere so it could be defined by x squared plus y squared plus Z squared is equal to 1 and what I'm going to focus on in this first video because it will take us several videos to do it it's just a parameterization of this surface right over here and as you'll see this is often the hardest parts it takes a little bit of visualization and then after that it's kind of mechanical but it can be kind of hairy at the same time so it's worth going through so first let's just think about how we can parameterize and I have trouble even saying the word how we can parameterize this unit sphere as a function of two parameters so let's think about it let's think about it a little bit so first let's just think about let's just think about the unit sphere I'm gonna take a side view of the unit sphere so let's take the unit sphere so this right over here is our z-axis that's our z-axis and then over here I'm going to draw this is going to be not just the X or the y axis this is going to be the entire XY plane viewed from the side that is the X the X Y plane now our sphere our sphere our unit sphere might look something like this the unit sphere itself is not too hard to visualize it might look something like that the radius let me make it very clear the radius at any point is 1 so this length right over here is 1 that length right over there is 1 and this is a sphere not just a circle so I could even shade it in a little bit just to make it clear that this thing has some dimensionality to it so that's shading it in it's kind of makes it look a little bit more spherical now let's attempt to parameterize this and there's a first step let's just think if we didn't have to think above and below the XY plane if we just thought about where this unit sphere intersected the XY plane how we could parameterize that so let's just think about it so where intersects the XY plane intersect there and there and actually everywhere so it intersects it right over there so let's just draw the XY plane and think about that intersection then we could think about what happens as we go above and below the XY plane so on the XY plane this little region where we where we just shade it in so let me draw so this is so now you could view this as almost a top view the z-axis is now going to be pointing straight out at you straight out of the screen so that's X let me draw it so that's X and then this right over there is Y so this thing that we were viewing sideways now we're viewing it from the top and so now our unit sphere is going to look is going to look something like this viewed from above and this what I just drew this dotted circle right over here this is going to be where our unit sphere intersects what I would label that Y that should be X don't want to confuse you already let me clear that so this is our x-axis this is our x-axis so this little dotted blue circle this is where our unit sphere intersects the XY plane and so using this we can start to think about how to parameterize at least our x and y values our x and y coordinates as a function of a first parameter so the first parameter we can think of something that is so this is the z-axis popping straight out at us so we're essentially if we're rotating around that z-axis viewed from above we could imagine an angle an angle and I'll call that angle I will call that angle s which is essentially saying how much we're rotating from the x axis towards the y axis in you could think about it in the XY plane or in a plane that is parallel to the XY plane or you could say going around the z axis z axis popping straight up at us and the radius here is always 1 it's a unit sphere so given this parameter s what would be your X&Y coordinate our thinking about it right if we're sitting in the XY plane well the x coordinate this goes goes back to the unit circle definition of our trig functions the x coordinate is going to be cosine of s it would be the radius which is 1 times the cosine of s and the y-coordinate would be one times the sine of s that's actually where we get our definitions for cosine and sine from so that's pretty straightforward and in this case Z is obviously equal to zero so if we wanted to add our Z coordinate here Z is 0 we are sitting we are sitting in the XY plane but now let's think about what happens if we go above and below the XY plane remember this is this is in any plane that is parallel to the XY plane this is saying how we are rotated around the z-axis now let's think about if we go above and below it and to figure out how far above or below it I'm going to introduce another parameter and this new parameter I'm going to introduce is T T is how much we have rotated above and below the XY plane now what's interesting about that is if we take any other cross-section that is parallel to the XY plane now we are going to have a smaller radius let me make that clear so if we if we if we're this if we're right over there if we're right over there now where this plane intersects our unit sphere the radius is smaller the radius is smaller than it was before well what would be this new radius well a little bit of trigonometry a little bit of trigonometry it's the same as it's the same as this length right over here which is going to be cosine so it's going to be cosine of T so the radius the radius is going to be cosine of T and it still works over here because if T goes all the way to 0 cosine of 0 is 1 and then that works right over there when we're in the XY plane so the radius the radius over here is going to be so that is right over there is cosine of 0 so this is when T this is when T is equal to zero and we haven't rotated above or below the XY plane but if we are if we have rotated above in the X Y above the XY plane the radius has changed it is now cosine of T and now we can use that to truly parameterize X&Y anywhere so now let's look at this cross-section so where you could you were not necessarily the XY plane we're in something that's parallel to the XY plane and so if we're up here now all of a sudden the cross section if we viewed it if we view it from above might look something like this it might look something something like this we're viewing it from above this cross-section right over here our radius our radius right over here is cosine cosine of T and so given that I guess altitude that we're at what would now be the parameterization using s of x and y well it's the exact same thing except now that I've written except now our radius isn't a fixed one it is now a function of T so we're now a little bit higher so now our our x-coordinate our x-coordinate is going to be our radius which is cosine of t cosine of t that's just our radius x times cosine of s times cosine of s how much we've angled around and in this case in this case s has gone all the way around here s has gone all the way around there so it's going to be cosine of T times cosine of s and then our y-coordinate is going to be our radius which is cosine of T times sine of s same exact logic here except now we have a different radius our radius is no longer 1 times sine of s sine of s running out of space let me scroll to the right a little bit and then and this looks very confusing but you said to say at any given level we are we're parallel to the x axis we're kind of tracing out another circle where another plane intersects our unit sphere we're now then rotating around with s and so our radius will change it's a function of how much above and below we've rotated how much above or below the xy-plane we've rotated so this is just our radius instead of 1 and then s is how much we've rotated around the z-axis same there for the y-coordinate and then the z coordinate is pretty straight forward it's going to be full it's going to be completely a function of T it's not dependent on how much we've rotated around here at any given at any given altitude it is what our altitude actually is and we can go straight to this diagram right over here our Z coordinate is just going to be the sine of T so there Z Z is equal to of T so let me write that down so Z is going to be equal to sine of T so now every point on the sphere sphere can be described as a function of T and s and we have to think about over what range will they be defined well s is going to go at any given level you could think for any given T s is going to go all the way around we see that right over here at any given level viewed from above s is going to go all the way around so thinking about it in radians s is going to be between 0 and 2pi and T is essentially our altitude in the Z direction so T can go all the way down here which would be negative PI over 2 so T can be between negative PI over 2 and it can go all the way up to PI over 2 it doesn't need to go all the way back down again and so goes all the way back I always goes up to PI over 2 and then we have our prior ammeter is a ssin let me write this down in a form that you might recognize even more if we wanted to write our surface as a position vector function we could write it like this we could write it R is a function of s and T and it is equal to our our X component our I component is going to be cosine of T cosine of T cosine of s cosine of s I and then plus our Y component is cosine of T cosine of T sine of s sine of s plus our Z component which is the sine which is just oh I forgot our J vector J plus the Z component which is just sine sine of T sine of T K and we're done and this is and these are the ranges that those parameters will take on so that's just the first step we've parameterised this surface now we're going to have this actually set up the surface integral it's going to involve a little bit of taking a cross product which can get hairy and then we can actually evaluate the integral itself