| align:start position:0% |
| |
| okay today I'm speaking about the first |
|
|
| align:start position:0% |
| okay today I'm speaking about the first |
| |
|
|
| align:start position:0% |
| okay today I'm speaking about the first |
| of the three great partial differential |
|
|
| align:start position:0% |
| of the three great partial differential |
| |
|
|
| align:start position:0% |
| of the three great partial differential |
| equation partial differential equations |
|
|
| align:start position:0% |
| equation partial differential equations |
| |
|
|
| align:start position:0% |
| equation partial differential equations |
| so this one is called Laplace's equation |
|
|
| align:start position:0% |
| so this one is called Laplace's equation |
| |
|
|
| align:start position:0% |
| so this one is called Laplace's equation |
| named after Laplace and you see partial |
|
|
| align:start position:0% |
| named after Laplace and you see partial |
| |
|
|
| align:start position:0% |
| named after Laplace and you see partial |
| derivatives so we have I don't have time |
|
|
| align:start position:0% |
| derivatives so we have I don't have time |
| |
|
|
| align:start position:0% |
| derivatives so we have I don't have time |
| this equation is in steady state I have |
|
|
| align:start position:0% |
| this equation is in steady state I have |
| |
|
|
| align:start position:0% |
| this equation is in steady state I have |
| x and y I'm in the XY plane and I have |
|
|
| align:start position:0% |
| x and y I'm in the XY plane and I have |
| |
|
|
| align:start position:0% |
| x and y I'm in the XY plane and I have |
| second derivatives in X and in Y so I'm |
|
|
| align:start position:0% |
| second derivatives in X and in Y so I'm |
| |
|
|
| align:start position:0% |
| second derivatives in X and in Y so I'm |
| looking for solutions to that equation |
|
|
| align:start position:0% |
| looking for solutions to that equation |
| |
|
|
| align:start position:0% |
| looking for solutions to that equation |
| and of course I'm given some boundary |
|
|
| align:start position:0% |
| and of course I'm given some boundary |
| |
|
|
| align:start position:0% |
| and of course I'm given some boundary |
| values |
|
|
| align:start position:0% |
| values |
| |
|
|
| align:start position:0% |
| values |
| so time is not here the boundary values |
|
|
| align:start position:0% |
| so time is not here the boundary values |
| |
|
|
| align:start position:0% |
| so time is not here the boundary values |
| the boundary is in the XY plane maybe a |
|
|
| align:start position:0% |
| the boundary is in the XY plane maybe a |
| |
|
|
| align:start position:0% |
| the boundary is in the XY plane maybe a |
| circle think about a circle in the XY |
|
|
| align:start position:0% |
| circle think about a circle in the XY |
| |
|
|
| align:start position:0% |
| circle think about a circle in the XY |
| plane and on the circle I know the |
|
|
| align:start position:0% |
| plane and on the circle I know the |
| |
|
|
| align:start position:0% |
| plane and on the circle I know the |
| solution U |
|
|
| align:start position:0% |
| solution U |
| |
|
|
| align:start position:0% |
| solution U |
| so the boundary values around the circle |
|
|
| align:start position:0% |
| so the boundary values around the circle |
| |
|
|
| align:start position:0% |
| so the boundary values around the circle |
| or give it and I have to find the |
|
|
| align:start position:0% |
| or give it and I have to find the |
| |
|
|
| align:start position:0% |
| or give it and I have to find the |
| temperature you inside the circle so I |
|
|
| align:start position:0% |
| temperature you inside the circle so I |
| |
|
|
| align:start position:0% |
| temperature you inside the circle so I |
| know the temperature on the boundary I |
|
|
| align:start position:0% |
| know the temperature on the boundary I |
| |
|
|
| align:start position:0% |
| know the temperature on the boundary I |
| let it settle down and I want to know |
|
|
| align:start position:0% |
| let it settle down and I want to know |
| |
|
|
| align:start position:0% |
| let it settle down and I want to know |
| the temperature inside and the beauty is |
|
|
| align:start position:0% |
| the temperature inside and the beauty is |
| |
|
|
| align:start position:0% |
| the temperature inside and the beauty is |
| it solves that |
|
|
| align:start position:0% |
| it solves that |
| |
|
|
| align:start position:0% |
| it solves that |
| basic partial differential equation so |
|
|
| align:start position:0% |
| basic partial differential equation so |
| |
|
|
| align:start position:0% |
| basic partial differential equation so |
| let's find some solutions they might not |
|
|
| align:start position:0% |
| let's find some solutions they might not |
| |
|
|
| align:start position:0% |
| let's find some solutions they might not |
| match the boundary values but we can use |
|
|
| align:start position:0% |
| match the boundary values but we can use |
| |
|
|
| align:start position:0% |
| match the boundary values but we can use |
| them so u equal constant certainly |
|
|
| align:start position:0% |
| them so u equal constant certainly |
| |
|
|
| align:start position:0% |
| them so u equal constant certainly |
| solves the equation u equal x the second |
|
|
| align:start position:0% |
| solves the equation u equal x the second |
| |
|
|
| align:start position:0% |
| solves the equation u equal x the second |
| derivatives will be 0 u equal Y here's a |
|
|
| align:start position:0% |
| derivatives will be 0 u equal Y here's a |
| |
|
|
| align:start position:0% |
| derivatives will be 0 u equal Y here's a |
| better one x squared minus y squared so |
|
|
| align:start position:0% |
| better one x squared minus y squared so |
| |
|
|
| align:start position:0% |
| better one x squared minus y squared so |
| the second derivative in the x-direction |
|
|
| align:start position:0% |
| the second derivative in the x-direction |
| |
|
|
| align:start position:0% |
| the second derivative in the x-direction |
| is 2 the second derivative in the |
|
|
| align:start position:0% |
| is 2 the second derivative in the |
| |
|
|
| align:start position:0% |
| is 2 the second derivative in the |
| y-direction is minus 2 so I have 2 minus |
|
|
| align:start position:0% |
| y-direction is minus 2 so I have 2 minus |
| |
|
|
| align:start position:0% |
| y-direction is minus 2 so I have 2 minus |
| 2 it solves the equation or this one the |
|
|
| align:start position:0% |
| 2 it solves the equation or this one the |
| |
|
|
| align:start position:0% |
| 2 it solves the equation or this one the |
| second derivative in X is 0 second |
|
|
| align:start position:0% |
| second derivative in X is 0 second |
| |
|
|
| align:start position:0% |
| second derivative in X is 0 second |
| derivative in Y is 0 those are simple |
|
|
| align:start position:0% |
| derivative in Y is 0 those are simple |
| |
|
|
| align:start position:0% |
| derivative in Y is 0 those are simple |
| solutions but those are only a few |
|
|
| align:start position:0% |
| solutions but those are only a few |
| |
|
|
| align:start position:0% |
| solutions but those are only a few |
| solutions and we need an infinite |
|
|
| align:start position:0% |
| solutions and we need an infinite |
| |
|
|
| align:start position:0% |
| solutions and we need an infinite |
| sequence because we're going to match |
|
|
| align:start position:0% |
| sequence because we're going to match |
| |
|
|
| align:start position:0% |
| sequence because we're going to match |
| initial |
|
|
| align:start position:0% |
| initial |
| |
|
|
| align:start position:0% |
| initial |
| conditions ok so is there a path pattern |
|
|
| align:start position:0% |
| conditions ok so is there a path pattern |
| |
|
|
| align:start position:0% |
| conditions ok so is there a path pattern |
| here so this is degree zero constant |
|
|
| align:start position:0% |
| here so this is degree zero constant |
| |
|
|
| align:start position:0% |
| here so this is degree zero constant |
| these are degree one linear these are |
|
|
| align:start position:0% |
| these are degree one linear these are |
| |
|
|
| align:start position:0% |
| these are degree one linear these are |
| degree two quadratic so I hope for two |
|
|
| align:start position:0% |
| degree two quadratic so I hope for two |
| |
|
|
| align:start position:0% |
| degree two quadratic so I hope for two |
| cubic ones and then I hope for two |
|
|
| align:start position:0% |
| cubic ones and then I hope for two |
| |
|
|
| align:start position:0% |
| cubic ones and then I hope for two |
| fourth degree ones and that's the |
|
|
| align:start position:0% |
| fourth degree ones and that's the |
| |
|
|
| align:start position:0% |
| fourth degree ones and that's the |
| pattern that's the pattern let me find |
|
|
| align:start position:0% |
| pattern that's the pattern let me find |
| |
|
|
| align:start position:0% |
| pattern that's the pattern let me find |
| let me spot the |
|
|
| align:start position:0% |
| let me spot the |
| |
|
|
| align:start position:0% |
| let me spot the |
| the cubic ones X cube if I start with X |
|
|
| align:start position:0% |
| the cubic ones X cube if I start with X |
| |
|
|
| align:start position:0% |
| the cubic ones X cube if I start with X |
| cube of course the second X derivative |
|
|
| align:start position:0% |
| cube of course the second X derivative |
| |
|
|
| align:start position:0% |
| cube of course the second X derivative |
| is probably 6 X |
|
|
| align:start position:0% |
| is probably 6 X |
| |
|
|
| align:start position:0% |
| is probably 6 X |
| so I need the second Y derivative to be |
|
|
| align:start position:0% |
| so I need the second Y derivative to be |
| |
|
|
| align:start position:0% |
| so I need the second Y derivative to be |
| minus 6x and I think minus 3x y squared |
|
|
| align:start position:0% |
| minus 6x and I think minus 3x y squared |
| |
|
|
| align:start position:0% |
| minus 6x and I think minus 3x y squared |
| does it mine the second derivative of in |
|
|
| align:start position:0% |
| does it mine the second derivative of in |
| |
|
|
| align:start position:0% |
| does it mine the second derivative of in |
| Y is 2 times the minus 3x is minus 6x |
|
|
| align:start position:0% |
| Y is 2 times the minus 3x is minus 6x |
| |
|
|
| align:start position:0% |
| Y is 2 times the minus 3x is minus 6x |
| cancels the 6 X from that's from the |
|
|
| align:start position:0% |
| cancels the 6 X from that's from the |
| |
|
|
| align:start position:0% |
| cancels the 6 X from that's from the |
| second derivative there and it works so |
|
|
| align:start position:0% |
| second derivative there and it works so |
| |
|
|
| align:start position:0% |
| second derivative there and it works so |
| that fits the pattern but what is the |
|
|
| align:start position:0% |
| that fits the pattern but what is the |
| |
|
|
| align:start position:0% |
| that fits the pattern but what is the |
| pattern ok here it is it's fantastic |
|
|
| align:start position:0% |
| pattern ok here it is it's fantastic |
| |
|
|
| align:start position:0% |
| pattern ok here it is it's fantastic |
| it's I |
|
|
| align:start position:0% |
| it's I |
| |
|
|
| align:start position:0% |
| it's I |
| get I get these |
|
|
| align:start position:0% |
| get I get these |
| |
|
|
| align:start position:0% |
| get I get these |
| crazy polynomials from taking X plus iy |
|
|
| align:start position:0% |
| crazy polynomials from taking X plus iy |
| |
|
|
| align:start position:0% |
| crazy polynomials from taking X plus iy |
| to the different powers here to the |
|
|
| align:start position:0% |
| to the different powers here to the |
| |
|
|
| align:start position:0% |
| to the different powers here to the |
| first power if n is 1 |
|
|
| align:start position:0% |
| first power if n is 1 |
| |
|
|
| align:start position:0% |
| first power if n is 1 |
| and I just have X plus iy and I take the |
|
|
| align:start position:0% |
| and I just have X plus iy and I take the |
| |
|
|
| align:start position:0% |
| and I just have X plus iy and I take the |
| real part that's X so I'll take a real |
|
|
| align:start position:0% |
| real part that's X so I'll take a real |
| |
|
|
| align:start position:0% |
| real part that's X so I'll take a real |
| part of this |
|
|
| align:start position:0% |
| part of this |
| |
|
|
| align:start position:0% |
| part of this |
| the real part of this when n is 1 the |
|
|
| align:start position:0% |
| the real part of this when n is 1 the |
| |
|
|
| align:start position:0% |
| the real part of this when n is 1 the |
| real part is X |
|
|
| align:start position:0% |
| real part is X |
| |
|
|
| align:start position:0% |
| real part is X |
| what about when n is 2 can you can you |
|
|
| align:start position:0% |
| what about when n is 2 can you can you |
| |
|
|
| align:start position:0% |
| what about when n is 2 can you can you |
| square that in your head so we have x |
|
|
| align:start position:0% |
| square that in your head so we have x |
| |
|
|
| align:start position:0% |
| square that in your head so we have x |
| squared and we have I squared Y squared |
|
|
| align:start position:0% |
| squared and we have I squared Y squared |
| |
|
|
| align:start position:0% |
| squared and we have I squared Y squared |
| I squared be minus 1 so I have x squared |
|
|
| align:start position:0% |
| I squared be minus 1 so I have x squared |
| |
|
|
| align:start position:0% |
| I squared be minus 1 so I have x squared |
| and I have minus y spread look the real |
|
|
| align:start position:0% |
| and I have minus y spread look the real |
| |
|
|
| align:start position:0% |
| and I have minus y spread look the real |
| part of this when n is 2 the real part |
|
|
| align:start position:0% |
| part of this when n is 2 the real part |
| |
|
|
| align:start position:0% |
| part of this when n is 2 the real part |
| of X plus I Y squared the real part is x |
|
|
| align:start position:0% |
| of X plus I Y squared the real part is x |
| |
|
|
| align:start position:0% |
| of X plus I Y squared the real part is x |
| squared minus y squared and the |
|
|
| align:start position:0% |
| squared minus y squared and the |
| |
|
|
| align:start position:0% |
| squared minus y squared and the |
| imaginary part |
|
|
| align:start position:0% |
| imaginary part |
| |
|
|
| align:start position:0% |
| imaginary part |
| was the 2i X Y so the imaginary part |
|
|
| align:start position:0% |
| was the 2i X Y so the imaginary part |
| |
|
|
| align:start position:0% |
| was the 2i X Y so the imaginary part |
| that multiplies I is the 2xy this is our |
|
|
| align:start position:0% |
| that multiplies I is the 2xy this is our |
| |
|
|
| align:start position:0% |
| that multiplies I is the 2xy this is our |
| pattern when n is 2 and when n is 3 I |
|
|
| align:start position:0% |
| pattern when n is 2 and when n is 3 I |
| |
|
|
| align:start position:0% |
| pattern when n is 2 and when n is 3 I |
| take X plus I Y cubed and that begins |
|
|
| align:start position:0% |
| take X plus I Y cubed and that begins |
| |
|
|
| align:start position:0% |
| take X plus I Y cubed and that begins |
| with X cube like that and then I think |
|
|
| align:start position:0% |
| with X cube like that and then I think |
| |
|
|
| align:start position:0% |
| with X cube like that and then I think |
| that the other real part would be a |
|
|
| align:start position:0% |
| that the other real part would be a |
| |
|
|
| align:start position:0% |
| that the other real part would be a |
| minus 3 XY squared I think you should |
|
|
| align:start position:0% |
| minus 3 XY squared I think you should |
| |
|
|
| align:start position:0% |
| minus 3 XY squared I think you should |
| check that and then there will be an |
|
|
| align:start position:0% |
| check that and then there will be an |
| |
|
|
| align:start position:0% |
| check that and then there will be an |
| imaginary part well I think I could |
|
|
| align:start position:0% |
| imaginary part well I think I could |
| |
|
|
| align:start position:0% |
| imaginary part well I think I could |
| figure out the imaginary part as I think |
|
|
| align:start position:0% |
| figure out the imaginary part as I think |
| |
|
|
| align:start position:0% |
| figure out the imaginary part as I think |
| maybe something like minus is something |
|
|
| align:start position:0% |
| maybe something like minus is something |
| |
|
|
| align:start position:0% |
| maybe something like minus is something |
| like |
|
|
| align:start position:0% |
| like |
| |
|
|
| align:start position:0% |
| like |
| - |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| yeah maybe maybe it's |
|
|
| align:start position:0% |
| yeah maybe maybe it's |
| |
|
|
| align:start position:0% |
| yeah maybe maybe it's |
| 3y x squared minus y cube something like |
|
|
| align:start position:0% |
| 3y x squared minus y cube something like |
| |
|
|
| align:start position:0% |
| 3y x squared minus y cube something like |
| that |
|
|
| align:start position:0% |
| that |
| |
|
|
| align:start position:0% |
| that |
| that would be the real part and that |
|
|
| align:start position:0% |
| that would be the real part and that |
| |
|
|
| align:start position:0% |
| that would be the real part and that |
| would be the imaginary part when n is 3 |
|
|
| align:start position:0% |
| would be the imaginary part when n is 3 |
| |
|
|
| align:start position:0% |
| would be the imaginary part when n is 3 |
| and wonderfully wonderfully it works for |
|
|
| align:start position:0% |
| and wonderfully wonderfully it works for |
| |
|
|
| align:start position:0% |
| and wonderfully wonderfully it works for |
| all |
|
|
| align:start position:0% |
| all |
| |
|
|
| align:start position:0% |
| all |
| powers |
|
|
| align:start position:0% |
| powers |
| |
|
|
| align:start position:0% |
| powers |
| exponents n |
|
|
| align:start position:0% |
| exponents n |
| |
|
|
| align:start position:0% |
| exponents n |
| so I have now a sort of pretty big |
|
|
| align:start position:0% |
| so I have now a sort of pretty big |
| |
|
|
| align:start position:0% |
| so I have now a sort of pretty big |
| family of solutions a list a double list |
|
|
| align:start position:0% |
| family of solutions a list a double list |
| |
|
|
| align:start position:0% |
| family of solutions a list a double list |
| really the real parts and the imaginary |
|
|
| align:start position:0% |
| really the real parts and the imaginary |
| |
|
|
| align:start position:0% |
| really the real parts and the imaginary |
| parts for every N so I can use those |
|
|
| align:start position:0% |
| parts for every N so I can use those |
| |
|
|
| align:start position:0% |
| parts for every N so I can use those |
| to solve my find the solution U which |
|
|
| align:start position:0% |
| to solve my find the solution U which |
| |
|
|
| align:start position:0% |
| to solve my find the solution U which |
| I'm looking for the the temperature |
|
|
| align:start position:0% |
| I'm looking for the the temperature |
| |
|
|
| align:start position:0% |
| I'm looking for the the temperature |
| inside the circle right now of course I |
|
|
| align:start position:0% |
| inside the circle right now of course I |
| |
|
|
| align:start position:0% |
| inside the circle right now of course I |
| have a linear equation so |
|
|
| align:start position:0% |
| have a linear equation so |
| |
|
|
| align:start position:0% |
| have a linear equation so |
| if I have several solutions I can |
|
|
| align:start position:0% |
| if I have several solutions I can |
| |
|
|
| align:start position:0% |
| if I have several solutions I can |
| combine them and I still have a solution |
|
|
| align:start position:0% |
| combine them and I still have a solution |
| |
|
|
| align:start position:0% |
| combine them and I still have a solution |
| x plus 7y will be a solution plus 11x |
|
|
| align:start position:0% |
| x plus 7y will be a solution plus 11x |
| |
|
|
| align:start position:0% |
| x plus 7y will be a solution plus 11x |
| squared minus y squared no problem Plus |
|
|
| align:start position:0% |
| squared minus y squared no problem Plus |
| |
|
|
| align:start position:0% |
| squared minus y squared no problem Plus |
| 56 times 2xy those are all solutions so |
|
|
| align:start position:0% |
| 56 times 2xy those are all solutions so |
| |
|
|
| align:start position:0% |
| 56 times 2xy those are all solutions so |
| I'm going to find a solution |
|
|
| align:start position:0% |
| I'm going to find a solution |
| |
|
|
| align:start position:0% |
| I'm going to find a solution |
| my final solution you will be a |
|
|
| align:start position:0% |
| my final solution you will be a |
| |
|
|
| align:start position:0% |
| my final solution you will be a |
| combination of this this this this this |
|
|
| align:start position:0% |
| combination of this this this this this |
| |
|
|
| align:start position:0% |
| combination of this this this this this |
| this this and all the others for higher |
|
|
| align:start position:0% |
| this this and all the others for higher |
| |
|
|
| align:start position:0% |
| this this and all the others for higher |
| n that's going to be my solution and I |
|
|
| align:start position:0% |
| n that's going to be my solution and I |
| |
|
|
| align:start position:0% |
| n that's going to be my solution and I |
| will need that infinite family see |
|
|
| align:start position:0% |
| will need that infinite family see |
| |
|
|
| align:start position:0% |
| will need that infinite family see |
| partial differential equations we move |
|
|
| align:start position:0% |
| partial differential equations we move |
| |
|
|
| align:start position:0% |
| partial differential equations we move |
| up to infinite family of solutions |
|
|
| align:start position:0% |
| up to infinite family of solutions |
| |
|
|
| align:start position:0% |
| up to infinite family of solutions |
| instead of just a couple of null |
|
|
| align:start position:0% |
| instead of just a couple of null |
| |
|
|
| align:start position:0% |
| instead of just a couple of null |
| solutions okay so let me take an example |
|
|
| align:start position:0% |
| solutions okay so let me take an example |
| |
|
|
| align:start position:0% |
| solutions okay so let me take an example |
| let me take an example oh |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| my we're taking the region to be a |
|
|
| align:start position:0% |
| my we're taking the region to be a |
| |
|
|
| align:start position:0% |
| my we're taking the region to be a |
| circle |
|
|
| align:start position:0% |
| circle |
| |
|
|
| align:start position:0% |
| circle |
| okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so in that circle |
|
|
| align:start position:0% |
| so in that circle |
| |
|
|
| align:start position:0% |
| so in that circle |
| I'm looking for the solution U of x and |
|
|
| align:start position:0% |
| I'm looking for the solution U of x and |
| |
|
|
| align:start position:0% |
| I'm looking for the solution U of x and |
| y and actually in a circle it's pretty |
|
|
| align:start position:0% |
| y and actually in a circle it's pretty |
| |
|
|
| align:start position:0% |
| y and actually in a circle it's pretty |
| natural to use polar coordinates instead |
|
|
| align:start position:0% |
| natural to use polar coordinates instead |
| |
|
|
| align:start position:0% |
| natural to use polar coordinates instead |
| of x and y inside a circle that that's |
|
|
| align:start position:0% |
| of x and y inside a circle that that's |
| |
|
|
| align:start position:0% |
| of x and y inside a circle that that's |
| inconvenient in the xy-plane it's |
|
|
| align:start position:0% |
| inconvenient in the xy-plane it's |
| |
|
|
| align:start position:0% |
| inconvenient in the xy-plane it's |
| equation is |
|
|
| align:start position:0% |
| equation is |
| |
|
|
| align:start position:0% |
| equation is |
| involves x equals square root of 1 minus |
|
|
| align:start position:0% |
| involves x equals square root of 1 minus |
| |
|
|
| align:start position:0% |
| involves x equals square root of 1 minus |
| y squared or something |
|
|
| align:start position:0% |
| y squared or something |
| |
|
|
| align:start position:0% |
| y squared or something |
| I'll switch to polar coordinates R and |
|
|
| align:start position:0% |
| I'll switch to polar coordinates R and |
| |
|
|
| align:start position:0% |
| I'll switch to polar coordinates R and |
| theta |
|
|
| align:start position:0% |
| theta |
| |
|
|
| align:start position:0% |
| theta |
| well you might say |
|
|
| align:start position:0% |
| well you might say |
| |
|
|
| align:start position:0% |
| well you might say |
| remember we had these nice family of |
|
|
| align:start position:0% |
| remember we had these nice family of |
| |
|
|
| align:start position:0% |
| remember we had these nice family of |
| solutions |
|
|
| align:start position:0% |
| solutions |
| |
|
|
| align:start position:0% |
| solutions |
| what is it still good in polar |
|
|
| align:start position:0% |
| what is it still good in polar |
| |
|
|
| align:start position:0% |
| what is it still good in polar |
| coordinates well the fact is it's even |
|
|
| align:start position:0% |
| coordinates well the fact is it's even |
| |
|
|
| align:start position:0% |
| coordinates well the fact is it's even |
| better so the solutions you will be the |
|
|
| align:start position:0% |
| better so the solutions you will be the |
| |
|
|
| align:start position:0% |
| better so the solutions you will be the |
| real part and the imaginary part now |
|
|
| align:start position:0% |
| real part and the imaginary part now |
| |
|
|
| align:start position:0% |
| real part and the imaginary part now |
| what is X plus I Y |
|
|
| align:start position:0% |
| what is X plus I Y |
| |
|
|
| align:start position:0% |
| what is X plus I Y |
| in |
|
|
| align:start position:0% |
| in |
| |
|
|
| align:start position:0% |
| in |
| R and theta well we all know X is R cos |
|
|
| align:start position:0% |
| R and theta well we all know X is R cos |
| |
|
|
| align:start position:0% |
| R and theta well we all know X is R cos |
| theta plus |
|
|
| align:start position:0% |
| theta plus |
| |
|
|
| align:start position:0% |
| theta plus |
| I R |
|
|
| align:start position:0% |
| I R |
| |
|
|
| align:start position:0% |
| I R |
| sine theta and |
|
|
| align:start position:0% |
| sine theta and |
| |
|
|
| align:start position:0% |
| sine theta and |
| that's R |
|
|
| align:start position:0% |
| that's R |
| |
|
|
| align:start position:0% |
| that's R |
| times cos theta plus I sine theta the |
|
|
| align:start position:0% |
| times cos theta plus I sine theta the |
| |
|
|
| align:start position:0% |
| times cos theta plus I sine theta the |
| one unforgettable complex |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| Euler's formula e to the I theta |
|
|
| align:start position:0% |
| Euler's formula e to the I theta |
| |
|
|
| align:start position:0% |
| Euler's formula e to the I theta |
| ok |
|
|
| align:start position:0% |
| ok |
| |
|
|
| align:start position:0% |
| ok |
| now I need its nth power the nth power |
|
|
| align:start position:0% |
| now I need its nth power the nth power |
| |
|
|
| align:start position:0% |
| now I need its nth power the nth power |
| of this is wonderful the real part and |
|
|
| align:start position:0% |
| of this is wonderful the real part and |
| |
|
|
| align:start position:0% |
| of this is wonderful the real part and |
| imaginary part of the nth power is R to |
|
|
| align:start position:0% |
| imaginary part of the nth power is R to |
| |
|
|
| align:start position:0% |
| imaginary part of the nth power is R to |
| the nth eetu the i N |
|
|
| align:start position:0% |
| the nth eetu the i N |
| |
|
|
| align:start position:0% |
| the nth eetu the i N |
| see that's my X plus iy to the enth much |
|
|
| align:start position:0% |
| see that's my X plus iy to the enth much |
| |
|
|
| align:start position:0% |
| see that's my X plus iy to the enth much |
| nicer in polar coordinates because I can |
|
|
| align:start position:0% |
| nicer in polar coordinates because I can |
| |
|
|
| align:start position:0% |
| nicer in polar coordinates because I can |
| take the real part and the imaginary |
|
|
| align:start position:0% |
| take the real part and the imaginary |
| |
|
|
| align:start position:0% |
| take the real part and the imaginary |
| part right away it's R to the n cos n |
|
|
| align:start position:0% |
| part right away it's R to the n cos n |
| |
|
|
| align:start position:0% |
| part right away it's R to the n cos n |
| theta and |
|
|
| align:start position:0% |
| theta and |
| |
|
|
| align:start position:0% |
| theta and |
| R to the N sine of theta |
|
|
| align:start position:0% |
| R to the N sine of theta |
| |
|
|
| align:start position:0% |
| R to the N sine of theta |
| these are my solutions my long list of |
|
|
| align:start position:0% |
| these are my solutions my long list of |
| |
|
|
| align:start position:0% |
| these are my solutions my long list of |
| solutions to Laplace's equation and it's |
|
|
| align:start position:0% |
| solutions to Laplace's equation and it's |
| |
|
|
| align:start position:0% |
| solutions to Laplace's equation and it's |
| some combination of those some my final |
|
|
| align:start position:0% |
| some combination of those some my final |
| |
|
|
| align:start position:0% |
| some combination of those some my final |
| thing is going to be some combination of |
|
|
| align:start position:0% |
| thing is going to be some combination of |
| |
|
|
| align:start position:0% |
| thing is going to be some combination of |
| those some combination may be |
|
|
| align:start position:0% |
| those some combination may be |
| |
|
|
| align:start position:0% |
| those some combination may be |
| coefficients a sub n sum I |
|
|
| align:start position:0% |
| coefficients a sub n sum I |
| |
|
|
| align:start position:0% |
| coefficients a sub n sum I |
| can use these |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and I can use these so maybe B sub n R |
|
|
| align:start position:0% |
| and I can use these so maybe B sub n R |
| |
|
|
| align:start position:0% |
| and I can use these so maybe B sub n R |
| to the N sine and theta |
|
|
| align:start position:0% |
| to the N sine and theta |
| |
|
|
| align:start position:0% |
| to the N sine and theta |
| you may wonder what I'm doing but what |
|
|
| align:start position:0% |
| you may wonder what I'm doing but what |
| |
|
|
| align:start position:0% |
| you may wonder what I'm doing but what |
| I'm achieving is to find the a big the |
|
|
| align:start position:0% |
| I'm achieving is to find the a big the |
| |
|
|
| align:start position:0% |
| I'm achieving is to find the a big the |
| general solution of Laplace's equation |
|
|
| align:start position:0% |
| general solution of Laplace's equation |
| |
|
|
| align:start position:0% |
| general solution of Laplace's equation |
| instead of two constants that we had for |
|
|
| align:start position:0% |
| instead of two constants that we had for |
| |
|
|
| align:start position:0% |
| instead of two constants that we had for |
| an ordinary differential equation as C 1 |
|
|
| align:start position:0% |
| an ordinary differential equation as C 1 |
| |
|
|
| align:start position:0% |
| an ordinary differential equation as C 1 |
| and a C 2 here I have these guys go from |
|
|
| align:start position:0% |
| and a C 2 here I have these guys go from |
| |
|
|
| align:start position:0% |
| and a C 2 here I have these guys go from |
| up to infinity and goes up to infinity |
|
|
| align:start position:0% |
| up to infinity and goes up to infinity |
| |
|
|
| align:start position:0% |
| up to infinity and goes up to infinity |
| so I have many solutions and any |
|
|
| align:start position:0% |
| so I have many solutions and any |
| |
|
|
| align:start position:0% |
| so I have many solutions and any |
| combination working so that's the |
|
|
| align:start position:0% |
| combination working so that's the |
| |
|
|
| align:start position:0% |
| combination working so that's the |
| general solution that's the general |
|
|
| align:start position:0% |
| general solution that's the general |
| |
|
|
| align:start position:0% |
| general solution that's the general |
| solution and I would have to match that |
|
|
| align:start position:0% |
| solution and I would have to match that |
| |
|
|
| align:start position:0% |
| solution and I would have to match that |
| now here's the final step and not simple |
|
|
| align:start position:0% |
| now here's the final step and not simple |
| |
|
|
| align:start position:0% |
| now here's the final step and not simple |
| not always simple I have to match this |
|
|
| align:start position:0% |
| not always simple I have to match this |
| |
|
|
| align:start position:0% |
| not always simple I have to match this |
| to the boundary conditions that's what |
|
|
| align:start position:0% |
| to the boundary conditions that's what |
| |
|
|
| align:start position:0% |
| to the boundary conditions that's what |
| will tell me the constants of course as |
|
|
| align:start position:0% |
| will tell me the constants of course as |
| |
|
|
| align:start position:0% |
| will tell me the constants of course as |
| usual C 1 and C 2 came from the matching |
|
|
| align:start position:0% |
| usual C 1 and C 2 came from the matching |
| |
|
|
| align:start position:0% |
| usual C 1 and C 2 came from the matching |
| the conditions now I don't have just C 1 |
|
|
| align:start position:0% |
| the conditions now I don't have just C 1 |
| |
|
|
| align:start position:0% |
| the conditions now I don't have just C 1 |
| and C 2 I have this infinite family of |
|
|
| align:start position:0% |
| and C 2 I have this infinite family of |
| |
|
|
| align:start position:0% |
| and C 2 I have this infinite family of |
| A's infinite family of bees and I have a |
|
|
| align:start position:0% |
| A's infinite family of bees and I have a |
| |
|
|
| align:start position:0% |
| A's infinite family of bees and I have a |
| lot more to match because on the |
|
|
| align:start position:0% |
| lot more to match because on the |
| |
|
|
| align:start position:0% |
| lot more to match because on the |
| boundary here I have to match |
|
|
| align:start position:0% |
| boundary here I have to match |
| |
|
|
| align:start position:0% |
| boundary here I have to match |
| u 0 which is given so I might be given |
|
|
| align:start position:0% |
| u 0 which is given so I might be given |
| |
|
|
| align:start position:0% |
| u 0 which is given so I might be given |
| suppose I was given the u0 equal the |
|
|
| align:start position:0% |
| suppose I was given the u0 equal the |
| |
|
|
| align:start position:0% |
| suppose I was given the u0 equal the |
| temperature was |
|
|
| align:start position:0% |
| temperature was |
| |
|
|
| align:start position:0% |
| temperature was |
| equal one on the top half and on the |
|
|
| align:start position:0% |
| equal one on the top half and on the |
| |
|
|
| align:start position:0% |
| equal one on the top half and on the |
| bottom half say the temperature is |
|
|
| align:start position:0% |
| bottom half say the temperature is |
| |
|
|
| align:start position:0% |
| bottom half say the temperature is |
| minus one |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| that's a typical problem I have a |
|
|
| align:start position:0% |
| that's a typical problem I have a |
| |
|
|
| align:start position:0% |
| that's a typical problem I have a |
| circular region a |
|
|
| align:start position:0% |
| circular region a |
| |
|
|
| align:start position:0% |
| circular region a |
| the top half is held at one temperature |
|
|
| align:start position:0% |
| the top half is held at one temperature |
| |
|
|
| align:start position:0% |
| the top half is held at one temperature |
| the lower half is held at a different |
|
|
| align:start position:0% |
| the lower half is held at a different |
| |
|
|
| align:start position:0% |
| the lower half is held at a different |
| temperature I reach equilibrium |
|
|
| align:start position:0% |
| temperature I reach equilibrium |
| |
|
|
| align:start position:0% |
| temperature I reach equilibrium |
| everybody knows that along that line |
|
|
| align:start position:0% |
| everybody knows that along that line |
| |
|
|
| align:start position:0% |
| everybody knows that along that line |
| probably the temperature would be zero |
|
|
| align:start position:0% |
| probably the temperature would be zero |
| |
|
|
| align:start position:0% |
| probably the temperature would be zero |
| by symmetry but what's the temperature |
|
|
| align:start position:0% |
| by symmetry but what's the temperature |
| |
|
|
| align:start position:0% |
| by symmetry but what's the temperature |
| they are halfway up knotti not so easy |
|
|
| align:start position:0% |
| they are halfway up knotti not so easy |
| |
|
|
| align:start position:0% |
| they are halfway up knotti not so easy |
| or anywhere in there well the answer is |
|
|
| align:start position:0% |
| or anywhere in there well the answer is |
| |
|
|
| align:start position:0% |
| or anywhere in there well the answer is |
| you in the middle U of R and theta |
|
|
| align:start position:0% |
| you in the middle U of R and theta |
| |
|
|
| align:start position:0% |
| you in the middle U of R and theta |
| inside is given by |
|
|
| align:start position:0% |
| inside is given by |
| |
|
|
| align:start position:0% |
| inside is given by |
| that formula |
|
|
| align:start position:0% |
| that formula |
| |
|
|
| align:start position:0% |
| that formula |
| that formula and again the a ends and |
|
|
| align:start position:0% |
| that formula and again the a ends and |
| |
|
|
| align:start position:0% |
| that formula and again the a ends and |
| the B ends come by matching the getting |
|
|
| align:start position:0% |
| the B ends come by matching the getting |
| |
|
|
| align:start position:0% |
| the B ends come by matching the getting |
| the right answer on the boundary well |
|
|
| align:start position:0% |
| the right answer on the boundary well |
| |
|
|
| align:start position:0% |
| the right answer on the boundary well |
| there's a big theory there how do I |
|
|
| align:start position:0% |
| there's a big theory there how do I |
| |
|
|
| align:start position:0% |
| there's a big theory there how do I |
| match these that's called a Fourier |
|
|
| align:start position:0% |
| match these that's called a Fourier |
| |
|
|
| align:start position:0% |
| match these that's called a Fourier |
| series that's called a Fourier series so |
|
|
| align:start position:0% |
| series that's called a Fourier series so |
| |
|
|
| align:start position:0% |
| series that's called a Fourier series so |
| I'm finding the coefficients for a |
|
|
| align:start position:0% |
| I'm finding the coefficients for a |
| |
|
|
| align:start position:0% |
| I'm finding the coefficients for a |
| Fourier series the A's and B's that |
|
|
| align:start position:0% |
| Fourier series the A's and B's that |
| |
|
|
| align:start position:0% |
| Fourier series the A's and B's that |
| match a function around the boundary |
|
|
| align:start position:0% |
| match a function around the boundary |
| |
|
|
| align:start position:0% |
| match a function around the boundary |
| and I could match any function and |
|
|
| align:start position:0% |
| and I could match any function and |
| |
|
|
| align:start position:0% |
| and I could match any function and |
| Fourier series is another entirely |
|
|
| align:start position:0% |
| Fourier series is another entirely |
| |
|
|
| align:start position:0% |
| Fourier series is another entirely |
| separate video this we've done the job |
|
|
| align:start position:0% |
| separate video this we've done the job |
| |
|
|
| align:start position:0% |
| separate video this we've done the job |
| with Laplace's equation in a circle |
|
|
| align:start position:0% |
| with Laplace's equation in a circle |
| |
|
|
| align:start position:0% |
| with Laplace's equation in a circle |
| we've reduced the problem to a Fourier |
|
|
| align:start position:0% |
| we've reduced the problem to a Fourier |
| |
|
|
| align:start position:0% |
| we've reduced the problem to a Fourier |
| series problem we have found the general |
|
|
| align:start position:0% |
| series problem we have found the general |
| |
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| align:start position:0% |
| series problem we have found the general |
| solution and then to match it to a |
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| align:start position:0% |
| solution and then to match it to a |
| |
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| align:start position:0% |
| solution and then to match it to a |
| specific given boundary value that's a |
|
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| align:start position:0% |
| specific given boundary value that's a |
| |
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| align:start position:0% |
| specific given boundary value that's a |
| Fourier series problem so I'll have to |
|
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| align:start position:0% |
| Fourier series problem so I'll have to |
| |
|
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| align:start position:0% |
| Fourier series problem so I'll have to |
| put that off to the Fourier series video |
|
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| align:start position:0% |
| put that off to the Fourier series video |
| |
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| align:start position:0% |
| put that off to the Fourier series video |
| thank you |
