Deriving the equation for a membrane. A PDE course.

[Knight2023]

I'm taking my first, though I'm an autodidact yet calling it a course wouldn't do a harm, of Partial Differential Equations from Petrovsky's book. And in the second example of the first chapter, he derives the equation for the membrane deformed by a force but in equilibrium, here are the images:

https://1drv.ms/f/c/eff33ac169bd871a/Ei ... w?e=5QrOqb

My doubt is how he got the area of element $\Delta S$ as $(1 +u'_{x_1}^2 + u'_{x_2}^2)^{1/2}$?

[Red_Herring]

Knight2023 в сообщении #1650287 писал(а):

My doubt is how he got the area of element

This is Calculus II. If you do not know this, you are not ready for PDE.

[Knight2023]

Red_Herring в сообщении #1650290 писал(а):

Knight2023 в сообщении #1650287 писал(а):

My doubt is how he got the area of element

This is Calculus II. If you do not know this, you are not ready for PDE.

I don't know why you have assumed that I do not know Calculus II (Multivariable Calculus). But to be honest it is not clicking my mind how is this derived, though it seems very similar to arc length formula (it has square root, it has derivatives).

Can you please refer which formula of Multivariable Calculus is used there?

[Red_Herring]

Knight2023 в сообщении #1650305 писал(а):

I don't know why you have assumed that I do not know Calculus II

Formula of the surface given by equation $z=z(x,y)$. And since you do not know this very basic formula I have all reasons to conclude that you do not know Calculus II.

[Knight2023]

Red_Herring в сообщении #1650314 писал(а):

Knight2023 в сообщении #1650305 писал(а):

I don't know why you have assumed that I do not know Calculus II

Formula of the surface given by equation $z=z(x,y)$. And since you do not know this very basic formula I have all reasons to conclude that you do not know Calculus II.

Okay, I think I have figured it out. The arc length of $\Delta S$ in the direction of $x_1$ is

$$ \sqrt{ 1+ u'_{x_1}^2} \Delta x_1$$

and that in the direction of $x_2$ is

$$ \sqrt{ 1+ u'_{x_2}^2} \Delta x_2$$

Approximating this curved element with a rectangle and noticing that $u'_{x_i}^2$ are very small (owing to the restrictions he has put) and therefore there cross terms can be ignored, we find the area of $\Delta S$ as the product of two those two sides

$$ \sqrt{ (1+ u'_{x_1}^2 )\times ( 1+ u'_{x_2}^2)} \Delta x_1 \Delta x_2$$

$$ \sqrt{1+ u'_{x_1}^2 +u'_{x_2}^2} \Delta x_1 \Delta x_2 $$

I must admit that it was because of your strict comment that I did not know Multivariable Calculus that I was able to figure it out. Thank you. I hope all the way through my self-course in PDE your comments will guide me.

[Red_Herring]

Knight2023 в сообщении #1650433 писал(а):

I hope all the way through my self-course in PDE

What is the purpose of your self-course? There are plenty of textbooks targeting different audiences.

[Knight2023]

Red_Herring в сообщении #1650440 писал(а):

Knight2023 в сообщении #1650433 писал(а):

I hope all the way through my self-course in PDE

What is the purpose of your self-course? There are plenty of textbooks targeting different audiences.

I'm trying to complete BSc. level Mathematics on my own. The nearest, in future, aim (as there are others too) is that Maths is my optional subject in UPSC CSE exam. In this first course I would complete only the following topics:

Family of surfaces in three dimensions and formulation of partial differential equations; Solution of quasilinear partial differential equations of the first order, Cauchy’s method of characteristics; Linear partial differential equations of the second order with constant coefficients, canonical form; Equation of a vibrating string, heat equation, Laplace equation and their solutions.

I'm pretty fond of (and fearful too) of Russian books. My experience in school with Irodov's Physics book was good and bad, and then a German told me that in 20th century USSR's system of education was totally different from that of Western (and I'm neither a Russian nor a Westerner). My current experience with Filipov's problem book is that it takes me more than half an hour to solve his second order ODEs.

Are you dubious of Petrovsky's book? Your suggestions and feedback are welcome, and I would be thankful if you could add your credentials (I mean your experience as a professor, or a researcher or a student).

[Red_Herring]

1. Please, look at Calculus II with a correct derivation of $dS$ which works without assumption $|\nabla u| \ll 1$.
2. Petrovskii's book is a classical book, beautifully written but it is obsolete. I would recommend to look at the particular University, Math major program and follow recommended textbook there.

[Knight2023]

Red_Herring в сообщении #1650454 писал(а):

1. Please, look at Calculus II with a correct derivation of $dS$ which works without assumption $|\nabla u| \ll 1$.
2. Petrovskii's book is a classical book, beautifully written but it is obsolete. I would recommend to look at the particular University, Math major program and follow recommended textbook there.

Actually, what you're pointing at is not in Multivariable Calculus but in Differential Geometry. I have looked it up and found it in a book on Differential Geometry. Nevertheless, let me give the proof for the sake of the thread:

Consider a differential element of any surface, and mark four point on it A, B, C ,D with coordinates as $u(x_1,x_2)$, $u(x_1+\Delta x_1, x_2)$, $u(x_1+\Delta x_1, x_2+\Delta x_2)$, $u(x_!, x_2+\Delta x_2)$ respectively.

Consider the vector $\vec{AD}$:

$$ \langle x_1, x_2+\Delta x_2, u(x_1, x_2+\Delta x_2) \rangle - \langle x_1, x_2, u(x_1,x_2) \rangle $$

$$ \vec{AD} = \langle 0, \Delta x_2, u'_{x_2} \Delta x_2 \rangle $$

Similarly,

$$ \vec{AB} = \langle \Delta x_1, 0, u'_{x_1} \Delta x_1 \rangle $$

The area of this surface element is approximated by considering it a parallelogram whose area is given by the magnitude of cross product of its side vectors:

$$ \big| \vec{AD} \times \vec{AB} \big| = \sqrt{1+ u'_{x_1}^2+u'_{x_2}^2 }\Delta x_1 \Delta x_2 $$

And yes, I have not studied Differential Geometry yet, will it grossly affect my studies of Partial Differential Equations?

[Red_Herring]

Knight2023 в сообщении #1650500 писал(а):

Actually, what you're pointing at is not in Multivariable Calculus but in Differential Geometry

No matter how you call it, normally it is derived in Calculus II during 2nd year. Prerequisites for PDE class usually are Calculus II (including differential and integral calculuses; the latter includes path and surface integrals and Green, Gauss and Stokes formulae without differential forms) and ODE.