Learn about Partial Derivatives Video

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Learn about Partial Derivatives

Learn about Partial Derivatives Let’s now expand our knowledge of calculus to the 3rd dimension. So what was the first one what does a function look like in three dimensions and actually we’ll go over the different types because you can have a line and three dimensions or kind of a curve in three dimensions you can have a surface, you could have a vector field. There are different types of representations we’ll see when we start working with three dimensions but I think the most intuitive and none of this are directly intuitive, I think you have to really be able to visualize but the most intuitive to me at least is a surface and three dimensions and eventually we can expand this into end dimensions but then it becomes very hard to visualize. So we had our traditional x and y axis before but now let’s give another dimension of height so let’s say that this is my x axis and I’ll draw the positive quadrant. That’s my x axis, that’s the y axis and that’s the z axis and the convention is to kind of follow the right hand rule where the x axis cross. Taking the cross product of the x axis with the y axis is equal to the z axis, what do I mean by that. This is x that is colorfully don’t go well together, this is y, this is z. What do I mean by the cross product so if this is the unit vector and the I and the x direction so that’s I and let’s say that is a link to 1. This is in the y direction so it’s j little cap and that cap just means it’s a unit vector and in the y direction it some has magnitude of 1 and I’ll do it and use a different color for z, z is up and the unit vector for there is k. So just and this is just the convention that i cross j = k and that’s just the convention for drawing the x axis do we make increasing x pointing out this way that we’d make increasing x .inwards and this gives us the convention so i goes to this direction, I’m trying to make sure I can do my hand properly. So let me draw the cross product, so if you take the first vector your index finger in the direction of the first vector, middle finger in the direction of the 2nd vector and then the other fingers can do it and you need to do. So this is going in the direction of i that is going in the direction of j in the y direction and then your palm of your thumb and then your thumb is going to open up in this direction. Your thumb is going to point out which is the direction of k. So that’s just good to know where that convention from does this is kind of called a right handed coordinated system but let’s get to the meat and potatoes. So how do we define a surface in three dimensions? Well we can define z = f (x,y) so let’s do that we could just add the notation, z = f (x,y) and all that means is that if I give you and x value and a y value, you get a z value and so I don’t know it looked. Let’s pick one, z = F(x + y) =x2+y. So how do we plot the points on this surface and I’ll show you actually computer generated surfaces that are far more professional looking than anything. I could possibly try to draw but let’s see if we were to take if we said f of I don’t know f (2,1) well that would mean x= 2 so it’s 22 + 1= 5 and if we have to plot that point of the surface and maybe I’ll graft this as one of the little bit we go along the x axis 2 so 1, 2 and we go along the y axis 1. So if we take this point this is x = 2, y = 1 and then we go and then we say z = 5 so we will go up here. We go up 5 units here and we will plot that point. And you would see if you keep doing that you would plot a surface, and let me clean this up a little bit. So the natural question that you might want to ask and let me say let me show a surface. I’m afraid that when I manipulate this graph it will slow down my computer and I’ll start by sounding like I’m melting but I’ll take that risk, just bear with me. So here is a surface I use using this nice graph and it’s actually free, I’ll give you the link for it but this surface right here this is I’ll actually show the graph of this and we’ll start taking the partial. Don’t worry about this wall and we’ll get to this in the second but this is a function of x and y, you could see this is the x axis, the y axis, the height is the z axis and this will probably really slow down my computer but you can actually rotate it. I don’t want to slow things too much down while I’m trying to do my screen capture but anyway I think you’ll understand where you picked an x point. You point a y point and then z this surface right here without this line intersecting it, this surface right here is a function of x and y. So the question is how do we apply calculus to surfaces because actually let me bring that thing out again because if you look at the surface if you were to pick any arbitrate point on the surface and say what is this slope of that surface. Well kind of has no meaning because you have to kind of pick a direction, if you said what is the slope of the tangent line. Any point on this graph actually has an infinite number of tangent lines. Think of it this way take a bowl or something that maybe like this and then take a toothpick and make that toothpick tangent to the bowl and you can see that on any point on the bowl, you can just rotate that too thick around, so you kind have to pick the orientation of that toothpick. So what we’re going to learn is when you take a derivative in three dimensions you have to specify the direction that you’re taking the derivative in, so and this is why I actually drew this wall here. This wall is the equation y = .3 so you can kind of view it along this wall, y is a constant, right so if we assume that y is constant and maybe we could take just the derivative with respect to x, so we would essentially take the slope of this curve right here and let’s figure out how to do it. So first of all what is the equation of this surface and I just pick to the one that they had on Wikipedia but the equation of that surface and I’m going to remove this now so I don’t sound like I’m melting. The equation of that surface and let me just clear out everything because it will probably need the extra space and we go back to the pen tool, the equation is z= x2 + xy + y2. So we said if we want to take the derivative, it’s hard to you can’t just say there is one derivative, we have to pick a direction, we have to hold everything else constant and take the derivative with respect to just one variable and that is called the partial derivative. I know it sounds fancy but you’ll see it’s actually no hard I’ve been taking a regular derivative; you just have to make sure you remember which variable is a variable and which one is a constant. So let’s say we want to hold y constant and we just way for any constant y how much does z change with respect to x then we take the partial derivative and this is the notation, the partial derivative and you can view as a D with the top curled, the partial derivative of z with respect to x it equals, all we do is we take this expression, we take the derivative of x and we just assume that y is some constant. So what’s the derivative of 2 xs with respect to x well it’s just 2x. What’s the derivative of xy with respect to x, well if y is just a number it’s just a constant, remember not taking an implicit derivative here, y is just the constant so if you have some constant times x the derivative of that is just the constant = y and then what’s the derivative of y2 with respect to x. Well we’re assuming y2 is a constant, it’s just a number, right. Y is just a number so the derivative of just a number with respect to x is just 0 so the derivative of that is 0. So the partial derivative of z with respect to x is 2x +y. Now what does that mean, well that means that if I were to let me actually let me give you a little notation before I show what that means. Another way to write this exact same thing is if we wrote that f of xy = to the same thing x2 + xy + y2 the partial of f with respect to x we convert it as this. The partial derivative of f with respect to x and still a function of x and y it still depends on what constant y are using is equal to 2x +y. Anyway I thought it’s nice to see that notation. Now what does this mean well what is this slope of z with respect to x and say when x is 1 actually let’s pick smaller numbers, when x = .2, y = .3, Well we could use this. f the partial derivative of f with respect to x at .2,.3 = 2(x), that’s point 4 + y that’s .3, so the slope of this function with respect to x at the (.2, .3)= .7. Let’s see if we can visualize that. So that wall represents the line y = .3 and we want the slope at x = .2 so this is x as .2 right here. So the rate which is height or the rate which is z is changing with respect to x is .7 so every time x increases 1, z will increase by .7 so the slope is a little but less in one. I think you see that right that is tangent right here is increasing with increasing values of x but a little bit less than 45 %. Anyway, I all out of time see you in the next video.

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Learn about Partial Derivatives

Let’s now expand our knowledge of calculus to the 3rd dimension. So what was the first one what does a function look like in three dimensions and actually we’ll go over the different types because you can have a line and three dimensions or kind of a curve in three dimensions you can have a surface, you could have a vector field. There are different types of representations we’ll see when we start working with three dimensions but I think the most intuitive and none of this are directly intuitive, I think you have to really be able to visualize but the most intuitive to me at least is a surface and three dimensions and eventually we can expand this into end dimensions but then it becomes very hard to visualize.

So we had our traditional x and y axis before but now let’s give another dimension of height so let’s say that this is my x axis and I’ll draw the positive quadrant. That’s my x axis, that’s the y axis and that’s the z axis and the convention is to kind of follow the right hand rule where the x axis cross. Taking the cross product of the x axis with the y axis is equal to the z axis, what do I mean by that.

This is x that is colorfully don’t go well together, this is y, this is z. What do I mean by the cross product so if this is the unit vector and the I and the x direction so that’s I and let’s say that is a link to 1. This is in the y direction so it’s j little cap and that cap just means it’s a unit vector and in the y direction it some has magnitude of 1 and I’ll do it and use a different color for z, z is up and the unit vector for there is k.

So just and this is just the convention that i cross j = k and that’s just the convention for drawing the x axis do we make increasing x pointing out this way that we’d make increasing x .inwards and this gives us the convention so i goes to this direction, I’m trying to make sure I can do my hand properly. So let me draw the cross product, so if you take the first vector your index finger in the direction of the first vector, middle finger in the direction of the 2nd vector and then the other fingers can do it and you need to do. So this is going in the direction of i that is going in the direction of j in the y direction and then your palm of your thumb and then your thumb is going to open up in this direction. Your thumb is going to point out which is the direction of k.

So that’s just good to know where that convention from does this is kind of called a right handed coordinated system but let’s get to the meat and potatoes. So how do we define a surface in three dimensions? Well we can define z = f (x,y) so let’s do that we could just add the notation, z = f (x,y) and all that means is that if I give you and x value and a y value, you get a z value and so I don’t know it looked. Let’s pick one, z = F(x + y) =x2+y. So how do we plot the points on this surface and I’ll show you actually computer generated surfaces that are far more professional looking than anything. I could possibly try to draw but let’s see if we were to take if we said f of I don’t know f (2,1) well that would mean x= 2 so it’s 22 + 1= 5 and if we have to plot that point of the surface and maybe I’ll graft this as one of the little bit we go along the x axis 2 so 1, 2 and we go along the y axis 1. So if we take this point this is x = 2, y = 1 and then we go and then we say z = 5 so we will go up here. We go up 5 units here and we will plot that point.

And you would see if you keep doing that you would plot a surface, and let me clean this up a little bit. So the natural question that you might want to ask and let me say let me show a surface. I’m afraid that when I manipulate this graph it will slow down my computer and I’ll start by sounding like I’m melting but I’ll take that risk, just bear with me.

So here is a surface I use using this nice graph and it’s actually free, I’ll give you the link for it but this surface right here this is I’ll actually show the graph of this and we’ll start taking the partial. Don’t worry about this wall and we’ll get to this in the second but this is a function of x and y, you could see this is the x axis, the y axis, the height is the z axis and this will probably really slow down my computer but you can actually rotate it.

I don’t want to slow things too much down while I’m trying to do my screen capture but anyway I think you’ll understand where you picked an x point. You point a y point and then z this surface right here without this line intersecting it, this surface right here is a function of x and y.

So the question is how do we apply calculus to surfaces because actually let me bring that thing out again because if you look at the surface if you were to pick any arbitrate point on the surface and say what is this slope of that surface. Well kind of has no meaning because you have to kind of pick a direction, if you said what is the slope of the tangent line. Any point on this graph actually has an infinite number of tangent lines. Think of it this way take a bowl or something that maybe like this and then take a toothpick and make that toothpick tangent to the bowl and you can see that on any point on the bowl, you can just rotate that too thick around, so you kind have to pick the orientation of that toothpick.

So what we’re going to learn is when you take a derivative in three dimensions you have to specify the direction that you’re taking the derivative in, so and this is why I actually drew this wall here. This wall is the equation y = .3 so you can kind of view it along this wall, y is a constant, right so if we assume that y is constant and maybe we could take just the derivative with respect to x, so we would essentially take the slope of this curve right here and let’s figure out how to do it.

So first of all what is the equation of this surface and I just pick to the one that they had on Wikipedia but the equation of that surface and I’m going to remove this now so I don’t sound like I’m melting. The equation of that surface and let me just clear out everything because it will probably need the extra space and we go back to the pen tool, the equation is z= x2 + xy + y2. So we said if we want to take the derivative, it’s hard to you can’t just say there is one derivative, we have to pick a direction, we have to hold everything else constant and take the derivative with respect to just one variable and that is called the partial derivative. I know it sounds fancy but you’ll see it’s actually no hard I’ve been taking a regular derivative; you just have to make sure you remember which variable is a variable and which one is a constant.

So let’s say we want to hold y constant and we just way for any constant y how much does z change with respect to x then we take the partial derivative and this is the notation, the partial derivative and you can view as a D with the top curled, the partial derivative of z with respect to x it equals, all we do is we take this expression, we take the derivative of x and we just assume that y is some constant. So what’s the derivative of 2 xs with respect to x well it’s just 2x.

What’s the derivative of xy with respect to x, well if y is just a number it’s just a constant, remember not taking an implicit derivative here, y is just the constant so if you have some constant times x the derivative of that is just the constant = y and then what’s the derivative of y2 with respect to x. Well we’re assuming y2 is a constant, it’s just a number, right. Y is just a number so the derivative of just a number with respect to x is just 0 so the derivative of that is 0. So the partial derivative of z with respect to x is 2x +y.

Now what does that mean, well that means that if I were to let me actually let me give you a little notation before I show what that means. Another way to write this exact same thing is if we wrote that f of xy = to the same thing x2 + xy + y2 the partial of f with respect to x we convert it as this. The partial derivative of f with respect to x and still a function of x and y it still depends on what constant y are using is equal to 2x +y.

Anyway I thought it’s nice to see that notation. Now what does this mean well what is this slope of z with respect to x and say when x is 1 actually let’s pick smaller numbers, when x = .2, y = .3, Well we could use this. f the partial derivative of f with respect to x at .2,.3 = 2(x), that’s point 4 + y that’s .3, so the slope of this function with respect to x at the (.2, .3)= .7. Let’s see if we can visualize that. So that wall represents the line y = .3 and we want the slope at x = .2 so this is x as .2 right here. So the rate which is height or the rate which is z is changing with respect to x is .7 so every time x increases 1, z will increase by .7 so the slope is a little but less in one. I think you see that right that is tangent right here is increasing with increasing values of x but a little bit less than 45 %.

Anyway, I all out of time see you in the next video.

Transcription by: Scribe4you Transcription Services