Looks absolutely awesome, interesting. When I have solving this equations, I use just something I call "partial integral" - just basic integral, but with arbitrary constant as function, independent on integration variable :D
I love PDEs and FPDEs = functional partial differential equations. That means the dependent variable, in this case, U, and its partial derivatives, evaluated not just at x and y, but at arbitrary functions of x and y!
dr peyam, could I ask you to make a video about the derivation and explanation of the 'triple product rule' for the partial derivatives of three interdependent variables? namely, why is this product equal to -1 and what does it mean?
it's used mainly in thermodynamics, but I believe that the idea is purely of mathematical nature. if we have two variables x and y, given implicitly as f(x, y) = 0, then (dx/dy)(dy/dx) = 1. and apparently, if we have three variables x, y and z, given implicitly as f(x, y, z) = 0, then (dx/dy)(dy/dz)(dz/dx) = -1, but it's also very common to see it written as dx/dy = - ((dz/dy) / (dz/dx)).
@@michalbotor The triple product rule, known variously as the cyclic chain rule, cyclic relation, or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables. The rule finds application in thermodynamics, where frequently three variables can be related by a function of the form f(x, y, z) = 0, so each variable is given as an implicit function of the other two variables. For example, an equation of state for a fluid relates temperature, pressure, and volume in this manner. The triple product rule for such interrelated variables x, y, and z comes from using a reciprocity relation on the result of the implicit function theorem in two variables and is given by (I suggest you copy and paste the following into a "markdown viewer" extension) \left(\frac{\partial x}{\partial y} ight)_z\left(\frac{\partial y}{\partial z} ight)_x\left(\frac{\partial z}{\partial x} ight)_y = -1. Note: The third variable is considered to be an implicit function of the other two. Here the subscripts indicate which variables are held constant when the partial derivative is taken. That is, to explicitly compute the partial derivative of x with respect to y with z held constant, one would write x as a function of y and z and take the partial derivative of this function with respect to y only. The advantage of the triple product rule is that by rearranging terms, one can derive a number of substitution identities which allow one to replace partial derivatives which are difficult to analytically evaluate, experimentally measure, or integrate with quotients of partial derivatives which are easier to work with. For example, \left(\frac{\partial x}{\partial y} ight)_z = - \frac{\left(\frac{\partial z}{\partial y} ight)_x}{\left(\frac{\partial z}{\partial x} ight)_y} Various other forms of the rule are present in the literature; these can be derived by permuting the variables {x, y, z}. Triple product rule The triple product rule, known variously as the cyclic chain rule, cyclic relation, or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables. The rule finds application in thermodynamics, where frequently three variables can be related by a function of the form f(x, y, z) = 0, so each variable is given as an implicit function of the other two variables. For example, an equation of state for a fluid relates temperature, pressure, and volume in this manner. The triple product rule for such interrelated variables x, y, and z comes from using a reciprocity relation on the result of the implicit function theorem in two variables and is given by \left(\frac{\partial x}{\partial y} ight)_z\left(\frac{\partial y}{\partial z} ight)_x\left(\frac{\partial z}{\partial x} ight)_y = -1. Note: The third variable is considered to be an implicit function of the other two. Additional recommended knowledge Weighing the Right Way Better Weighing Performance In 6 Easy Steps 8 Steps to a Clean Balance - and 5 Solutions to Keep It Clean Here the subscripts indicate which variables are held constant when the partial derivative is taken. That is, to explicitly compute the partial derivative of x with respect to y with z held constant, one would write x as a function of y and z and take the partial derivative of this function with respect to y only. The advantage of the triple product rule is that by rearranging terms, one can derive a number of substitution identities which allow one to replace partial derivatives which are difficult to analytically evaluate, experimentally measure, or integrate with quotients of partial derivatives which are easier to work with. For example, \left(\frac{\partial x}{\partial y} ight)_z = - \frac{\left(\frac{\partial z}{\partial y} ight)_x}{\left(\frac{\partial z}{\partial x} ight)_y} Various other forms of the rule are present in the literature; these can be derived by permuting the variables {x, y, z}. Derivation An informal derivation follows. Suppose that f(x, y, z) = 0. Write z as a function of x and y. Thus the total derivative dz is dz = \left(\frac{\partial z}{\partial x} ight)_y dx + \left(\frac{\partial z}{\partial y} ight)_x dy Suppose that we move along a curve with dz = 0, where the curve is parameterized by x. Thus y can be written in terms of x, so on this curve dy = \left(\frac{\partial y}{\partial x} ight)_z dx Therefore the equation for dz = 0 becomes 0 = \left(\frac{\partial z}{\partial x} ight)_y dx + \left(\frac{\partial z}{\partial y} ight)_x \left(\frac{\partial y}{\partial x} ight)_z dx Dividing by dx and rearranging terms gives \left(\frac{\partial z}{\partial x} ight)_y = -\left(\frac{\partial z}{\partial y} ight)_x \left(\frac{\partial y}{\partial x} ight)_z Dividing by the derivatives on the right hand side gives the triple product rule \left(\frac{\partial x}{\partial y} ight)_z\left(\frac{\partial y}{\partial z} ight)_x\left(\frac{\partial z}{\partial x} ight)_y = -1 Note that this proof makes many implicit assumptions regarding the existence of partial derivatives, the existence of the total derivative dz, the ability to construct a curve in some neighborhood with dz = 0, and the nonzero value of partial derivatives and their reciprocals. A formal proof based on mathematical analysis would eliminate these potential ambiguities and grey zones. TL;DR www.chemeurope.com/en/encyclopedia/Triple_product_rule.html
4:56 Doesn't u_{xy} = (u_x)_y? Yes, for reasonably nice functions, mixed partial derivatives are equal, but (to quote the Gershwins) it ain't necessarily so.
@@drpeyam Agreed! Still, every introduction to partial derivatives I've seen shows counterexamples. Of course, since this is about introducing PDEs, one could assume that the audience already knows this fact. I probably would have noted it in passing, but then again I've never made a video in my life, so what do I know. Heck, I don't even know how to make a UA-cam alias! :D Good video as always.
PDE is extremely harder than that. Some PDE you can got solve via fourier transforms, fourier series. To prove somethings related to PDE you need a good knowledge in functional analysis.
Have you ever solved delay differential equations with a second member is a linear function plus a function which depends only on t? That's quite hard Sir
In the context of Calculus, it means a partial derivative with respect to that variable. For instance, Uxx means d²U/dx² (can't write the del sign here). Uxy means d²U/dydx (note the order changes)
1. U_x =0 U= f(y) 2. U_xx =0 U= x*f(y)+g(y) 3. U_xx +U=0 Uhh, if this were an ODE then U= c1*e^(ix) +c2*e^(-ix) 4. U_xy=0 U= c1*x +c2*y +c3 These are my guess, I've only taken Diff EQ 1 so I'm working with just basic PDE intuition.
I'm crying!!! Thank you soooo much for explaining the very basics. Every book and video expects you to already know this.
Looks absolutely awesome, interesting. When I have solving this equations, I use just something I call "partial integral" - just basic integral, but with arbitrary constant as function, independent on integration variable :D
i absolutely love his energy towards any topic, its uplifting!
He got his PhD by that
Hidden, but beautiful video on the PDE series!
math ops👀
I believe that partial differential equations are the first type of math that is considered "advanced" math at university.
2:12 The most interesting mathematician in the world, lol
history buff03
Lolllll best comment!
Did he speak in spanish? haha
@@blackpenredpen bro how come you have got just 4 likes?
Dr. Peyam, your smile is infectious. Great lecture, thank you for your content!!
Thanks, this gave me enough confidence to learn about PDEs more deeply
We get many of these in fluid dynamics.
Let u, ux, uxx, and uxy all equal 0. Done.
Jon Snow Yes... set them all generally equal to 0
😁
@Jon Snow that was a joke😂😂
ODEs were fun but I cant wait to learn PDEs! Thank you Dr. Peyam!
Dr.Peyam you should totally make videos on energy/max principles for PDEs!!!
Oh i just checked, youve already made some!
Yes, I can solve all 7 because I work as a telemarketer.
I love PDEs and FPDEs = functional partial differential equations. That means the dependent variable, in this case, U, and its partial derivatives, evaluated not just at x and y, but at arbitrary functions of x and y!
Thank you for making it simple
The general solution to the first PDE is actually f(y) + c, because it could also include a constant that doesn't depend on either x or y.
The c is part of f(y)
"Pulls out change of coordinates," "Aw shit here we go again."
dr peyam, could I ask you to make a video about the derivation and explanation of the 'triple product rule' for the partial derivatives of three interdependent variables? namely, why is this product equal to -1 and what does it mean?
Never heard about it
it's used mainly in thermodynamics, but I believe that the idea is purely of mathematical nature.
if we have two variables x and y, given implicitly as f(x, y) = 0, then (dx/dy)(dy/dx) = 1.
and apparently, if we have three variables x, y and z, given implicitly as f(x, y, z) = 0, then (dx/dy)(dy/dz)(dz/dx) = -1, but it's also very common to see it written as dx/dy = - ((dz/dy) / (dz/dx)).
@@michalbotor The triple product rule, known variously as the cyclic chain rule, cyclic relation, or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables. The rule finds application in thermodynamics, where frequently three variables can be related by a function of the form f(x, y, z) = 0, so each variable is given as an implicit function of the other two variables. For example, an equation of state for a fluid relates temperature, pressure, and volume in this manner. The triple product rule for such interrelated variables x, y, and z comes from using a reciprocity relation on the result of the implicit function theorem in two variables and is given by (I suggest you copy and paste the following into a "markdown viewer" extension)
\left(\frac{\partial x}{\partial y}
ight)_z\left(\frac{\partial y}{\partial z}
ight)_x\left(\frac{\partial z}{\partial x}
ight)_y = -1.
Note: The third variable is considered to be an implicit function of the other two.
Here the subscripts indicate which variables are held constant when the partial derivative is taken. That is, to explicitly compute the partial derivative of x with respect to y with z held constant, one would write x as a function of y and z and take the partial derivative of this function with respect to y only.
The advantage of the triple product rule is that by rearranging terms, one can derive a number of substitution identities which allow one to replace partial derivatives which are difficult to analytically evaluate, experimentally measure, or integrate with quotients of partial derivatives which are easier to work with. For example,
\left(\frac{\partial x}{\partial y}
ight)_z = - \frac{\left(\frac{\partial z}{\partial y}
ight)_x}{\left(\frac{\partial z}{\partial x}
ight)_y}
Various other forms of the rule are present in the literature; these can be derived by permuting the variables {x, y, z}.
Triple product rule
The triple product rule, known variously as the cyclic chain rule, cyclic relation, or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables. The rule finds application in thermodynamics, where frequently three variables can be related by a function of the form f(x, y, z) = 0, so each variable is given as an implicit function of the other two variables. For example, an equation of state for a fluid relates temperature, pressure, and volume in this manner. The triple product rule for such interrelated variables x, y, and z comes from using a reciprocity relation on the result of the implicit function theorem in two variables and is given by
\left(\frac{\partial x}{\partial y}
ight)_z\left(\frac{\partial y}{\partial z}
ight)_x\left(\frac{\partial z}{\partial x}
ight)_y = -1.
Note: The third variable is considered to be an implicit function of the other two.
Additional recommended knowledge
Weighing the Right Way
Better Weighing Performance In 6 Easy Steps
8 Steps to a Clean Balance - and 5 Solutions to Keep It Clean
Here the subscripts indicate which variables are held constant when the partial derivative is taken. That is, to explicitly compute the partial derivative of x with respect to y with z held constant, one would write x as a function of y and z and take the partial derivative of this function with respect to y only.
The advantage of the triple product rule is that by rearranging terms, one can derive a number of substitution identities which allow one to replace partial derivatives which are difficult to analytically evaluate, experimentally measure, or integrate with quotients of partial derivatives which are easier to work with. For example,
\left(\frac{\partial x}{\partial y}
ight)_z = - \frac{\left(\frac{\partial z}{\partial y}
ight)_x}{\left(\frac{\partial z}{\partial x}
ight)_y}
Various other forms of the rule are present in the literature; these can be derived by permuting the variables {x, y, z}.
Derivation
An informal derivation follows. Suppose that f(x, y, z) = 0. Write z as a function of x and y. Thus the total derivative dz is
dz = \left(\frac{\partial z}{\partial x}
ight)_y dx + \left(\frac{\partial z}{\partial y}
ight)_x dy
Suppose that we move along a curve with dz = 0, where the curve is parameterized by x. Thus y can be written in terms of x, so on this curve
dy = \left(\frac{\partial y}{\partial x}
ight)_z dx
Therefore the equation for dz = 0 becomes
0 = \left(\frac{\partial z}{\partial x}
ight)_y dx + \left(\frac{\partial z}{\partial y}
ight)_x \left(\frac{\partial y}{\partial x}
ight)_z dx
Dividing by dx and rearranging terms gives
\left(\frac{\partial z}{\partial x}
ight)_y = -\left(\frac{\partial z}{\partial y}
ight)_x \left(\frac{\partial y}{\partial x}
ight)_z
Dividing by the derivatives on the right hand side gives the triple product rule
\left(\frac{\partial x}{\partial y}
ight)_z\left(\frac{\partial y}{\partial z}
ight)_x\left(\frac{\partial z}{\partial x}
ight)_y = -1
Note that this proof makes many implicit assumptions regarding the existence of partial derivatives, the existence of the total derivative dz, the ability to construct a curve in some neighborhood with dz = 0, and the nonzero value of partial derivatives and their reciprocals. A formal proof based on mathematical analysis would eliminate these potential ambiguities and grey zones.
TL;DR
www.chemeurope.com/en/encyclopedia/Triple_product_rule.html
Next can you please do Cauchy's Integral Formula but using matrices instead of scalars to determine the resolvent.
Great video as always!
Great as always.
4:56 Doesn't u_{xy} = (u_x)_y? Yes, for reasonably nice functions, mixed partial derivatives are equal, but (to quote the Gershwins) it ain't necessarily so.
Yeah, but you would have lots of trouble studying PDEs without that assumption
@@drpeyam Agreed! Still, every introduction to partial derivatives I've seen shows counterexamples. Of course, since this is about introducing PDEs, one could assume that the audience already knows this fact. I probably would have noted it in passing, but then again I've never made a video in my life, so what do I know. Heck, I don't even know how to make a UA-cam alias! :D Good video as always.
4:57 Uxy means first differentiate from x and then from y or in other way?
Doesn’t matter since uxy = uyx
thanks really needed this
Took me a second to get your t-shirt.
PDE is extremely harder than that. Some PDE you can got solve via fourier transforms, fourier series. To prove somethings related to PDE you need a good knowledge in functional analysis.
Great video! 😊
Nice video!
6:03 "Well, yes, but, actually, no"
Hi! Could you show us how to solve wave equations with this technique? And how to obtain general solutions like Bessel equations... please! 🙏
Wave equation ua-cam.com/video/KFS_Fs1ZGRw/v-deo.html
Have you ever solved delay differential equations with a second member is a linear function plus a function which depends only on t?
That's quite hard Sir
Not really
Dr Peyam, what is the meaning of a letter or number used as a subscript in math?
In the context of Calculus, it means a partial derivative with respect to that variable. For instance, Uxx means d²U/dx² (can't write the del sign here). Uxy means d²U/dydx (note the order changes)
Here u_x means differentiate u with respect to x, it’s a partial derivative
I don't know what a partial derivative is 😂
@@HawluchaMCPE partial derivative means you treat all other variables like a constant
@@nathanisbored oh I got it, I understand what it is but never knew the terminology. Thanks.
Hey Dr Peyam. Can you solve this problem ? ==> lim x --> 0+ (arctan(x+a/x) + bx + c)/x^5
Find all possible values of the triple (a, b, c) of constants
Do it in 2 cases: If a = 0 (in that case Taylor expand arctan) and if a is nonzero, in which case arctan becomes pi/2 or -pi/2
@@drpeyam Ok I,ll try. Thank you 😊
A jobb kezeddel is tudc írni?
Welcome to the balding club, Dr π🍖😀
Yup
nice
1. U_x =0
U= f(y)
2. U_xx =0
U= x*f(y)+g(y)
3. U_xx +U=0
Uhh, if this were an ODE then
U= c1*e^(ix) +c2*e^(-ix)
4. U_xy=0
U= c1*x +c2*y +c3
These are my guess, I've only taken Diff EQ 1 so I'm working with just basic PDE intuition.
Not bad, but watch the video
Aaaaa I forgot that the constants could be functions of y in 3. My guess for 4 was an alright one, but not completely correct.
Also forgot that
c*e^(ix)= c1*cos(x) +c2*sin(x)
@@drpeyam haha yeah, I did after posting. You gave good explanations on why the solutions are the way they are.
👆👆😂😂 can you give your white board...😂🤣🤸♀️😜😜
what if it is U_XX+U_YY=0
That’s Laplace’s equation
The canonical form is a but easier
Any one pde expert plz
uxx + u = 0 and uxx = 0... u = 0. What am I missing here?
There are other solutions
You just assumed Uxx = 0
@@gnikola2013 that was given in the thumbnail
@@GuyMichaely the thing is, what was given in the thumbnail were 4 different equations, not a system!