Intro to Partial Derivatives

Mark Misin
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Aerospace & Robotics Engineer
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INTUITION MATTERS! - Applied Calculus for Engineers-Complete

Calculus + Engineering + PID: Functions, Limits, Derivatives, Vectors, differential equations, integrals: BEST CALCULUS

34:53:37 of on-demand video • Updated October 2020

  • You will develop very strong intuition & understanding in Calculus
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English [Auto] Welcome back. So starting from this lecture we're going to start covering multi-variable calculus and the first topic that we're going to cover in multivariable calculus is going to be partial derivatives. So what is a partial derivative. Well let's remember that we start with discours with an airplane example where you had some kind of airplane flying in some kind of action and its speed was 800 kilometers per hour. And then from that from that we concluded that we had a function since the speed was constant. We had a function where we could relate distance with time. So we are a function was distance equals 800 times time. So that was our function and the graph of our function was a straight line graph and with a function of the equals 800 T. So that would be the axis for time and that would be the axis for distance and then later on when we learned about the derivative then we learned that it was very easy to take the derivative of this function. So we simply apply the operator d d t which means change with respect to time instantaneous change with respect to time. And then we apply it to D and then the same operator went to the other side of the equation and then you had 880 here. And then when you apply the operator in this case you would have eight hundred kilometers per hour. In fact you would get speed if you take the derivative of distance with respect to time. So we had a function where you had only one input which was t. Then you had your function 800 t that you went here and then you got your output the OK. However in real life outputs really depend only on one input. In real life there are outputs that depend on many inputs. And let me give you an easy example say that the our airplanes distance not only depends on time but also for example on the altitude. If you remember in one video I explain that the higher you go the lower the air density. And when you are somewhere high in the sky and flying there then you have less air density which means that in one cubic meter you have less air molecules. So let's say at 10 kilometers you would have less air there than at the sea level. So that would mean that you would have less air resistance there and you would cover more distance. So I'm just going to give you a multivariable function and I don't know if it's true or not. However that's not the purpose of it. I just want to show you what a partial truth is. So this equation might not make sense in real life but it is just for illustrative purposes. So I'm going to say that the airplane's distance depends not only on time it's not only a function of time but also altitude. So I could say that OK my distance equals 800 te plus 20 age and age here is alt.. So what that really means is that now I have two independent variables two inputs T and H which is altitude time and altitude. Then I have my function here 800 T plus 20 h. And then my output would be distance. OK so now as I have this function I would like to know how distance changes only with respect to time or only with respect to altitude because you see now when you have more than one input then you can use this operator anymore because this operator is one for one variable. Only when you have more independent variables that your distance depends on then you have to specify whether you want to know how distance changes with time or how distance changes with age. And for that there is another operator not the. And the remember the operator here was D and D. And here you would put the variable with respect to which you want to find the change. And here you put the function right here you put the function and then here you put your variable. So if you have more than one input then the operator that you would use would be a partial derivative operator and it would look like this. And then down here you're going to put the variable with respect to which you want to find the change. And here you put the function. So if I want to find how my distance changes with respect to time I will very simply write it like this. I'm still going to call it D. But this is a partial derivative the now and every time I write it like this it means Parshall's wrote. So every time I use this operator I use a partial derivative. So here I want to know how my distance changes with time around. And now a partial derivative with one variable means that I consider all other variables as a constant. So if I want to know how my distance change with respect time then it will look like this partial Delta respect t. You have a hundred t plots right here. Partial narrative with respect to t and then I would have 20 h here and again a partial derivatives with respect to one variable in this case. He means that I consider all other variables in this case h as a constant. That means that this term here will become zero and my change of distance with respect to time would be just 800. In this case again it would be eight hundred kilometers per hour. However I could choose to find out how my distance changes with respect to altitude not with respect to time. In this case I simply change this variable down here so I'm going to write partial D with respect to alt.. And that would mean that now I consider this part when I find the derivative and I consider this T part as a constant. So t Now to me is constant. So if I write something like this this 800 t with respect to h Plus this 20 h with respect to H then this will become zero and I will get 20 and the 20 is in fact the. It's a unitless number because distance has a unit of time. Right distance has kilometers but also the altitude has kilometers. So in fact in terms of units the units will cancel out right. It's like climbing trees or kilometers. So it's like having one here. So in fact this thing represents how your distance changes with respect to altitude. What it really means is if I change my altitude a little bit then how much will my distance change because of that. So the logical route is the same it's just now I specify with respect to which variable and taking the derivative and not is a very important difference. What if my function which is now the equals 800 T plus 20 h. What if I apply this operator. What if I say I want to know how much my distance changes with respect time. But I take this normal the this. This derivative operator or differentiation operator that we had always used before. What would be the difference then. Well if I now do this d d t 800 T plus d d t 20 h Dan. Now this term here will not be zero. So now I would have 800 here this term would be 800. And this term would be 20 times the h d t and OK this term would be zero if the altitude does not change with time. Then it would be zero. However if the altitude itself is a function of time if the altitude does change with respect to time then then it would not be zero. And I would have to find this term here. I would have to find how my health is changes with respect to time. So I would have to consider that I would have to find out if my altitude is a function of time or not. And if it's not then it would be zero. But if it is a function of time then I would need to know how my altitude changes with respect to time. However if I use the parts of the operator then I don't do that. I just say that. OK. I want to know how my distance changes with respect to time. And then I will get a hundred and then I'll just treat my age as a zero. And the same thing if I use the other operator the D H then here I would have to know now how my 800 how my time changes with respect to altitude. So think about how my time changes with respect to altitude. And then plus 20. Now this turn of course doesn't make sense. How can time change with respect to altitude. So if I change my altitude then then my time should change. OK. Obviously in real life it doesn't make sense. There should be no connection so this term should be zero. However if you use this operator then you would have to consider that you have to think about OK. Is time as a function of altitude. So does time change with respect to altitude at least formally or officially you would have to consider that. And if not then of course it would be zero. However if you use this operator here then you automatically consider all the derivatives with other variables will become zero and you will simply have partial Delta and the partial Delta h equals 20. And that's why in partial derivatives they're using another operator just to make that distinction so that if you're taking a derivative with respect to altitude then all the other inputs that you have in your function. All the other inputs you treat them as constants. And if you take a derivative of a constant then it will be zero. So that's a very important distinction.