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June 22nd, 2017, 11:59 AM   #11
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Yes, I was confused about it for a while too. But I thought mathematician was right in some way, because I am the noob.

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A function of several variables, say w= f(x, y, z), has "partial derivatives" because its graph would NOT be a curve in the xy-plane but a three dimensional surface in the four dimensional xyzw space. At each $\displaystyle (x_0, y_0, z_0, f(x_0, y_0, z_0))$ point, we could move in three different directions parallel to the three x, y, and z coordinate axes. There may be a different slope in each of those directions, the partial derivatives $\displaystyle f_x(x_0, y_0, z_0)$, $\displaystyle f_y(x_0, y_0, z_0)$, and $\displaystyle f_z(x_0, y_0, z_0)$. An equation that involves the derivatives of a function of several variables is a "partial differential equation".

Because our universe is a "four dimensional space-time continuum", physics problems tend to involve "partial differential equations" with independent variables, x, y, z, and t. Most famously, the "wave equation"
$\displaystyle \frac{\partial^2 \phi}{\partial x^2}+ \frac{\partial^2 \phi}{\partial y^2}+ \frac{\partial^2 \phi}{\partial z^2}= \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2}$
and the "diffusion" or "heat equation"
$\displaystyle \frac{\partial^2 \phi}{\partial x^2}+ \frac{\partial^2 \phi}{\partial y^2}+ \frac{\partial^2 \phi}{\partial z^2}= \kappa \frac{\partial \phi}{\partial t}$.
Anyway this makes a lot of sense to me now. Thanks.

Last edited by skipjack; July 18th, 2017 at 02:24 PM.
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July 18th, 2017, 10:11 AM   #12
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Originally Posted by MATHEMATICIAN View Post
examples:
None of those is an example of a partial differential equation.
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July 18th, 2017, 12:18 PM   #13
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A few notes on differential equations might help, including showing the difference between ordinary differential equations (ODEs) and partial differential equations (PDEs).

ODEs use the Roman letter d to denote the derivatives
PDEs use the stylized Greek letter $\displaystyle \partial $ to denote the derivatives.

ODEs have one independent and one dependent variable.
PDEs have one dependent and 2 or more independent variables.

A solution of either sort of differential equation is an expression or relationship between the variables, free of derivatives.

For both, the order of the equation is the order of highest derivative present.

For both, the degree of the equation is the degree of the highest order derivative (see example)

So


$\displaystyle \frac{{dy}}{{dx}} = x$ is a first order first degree ODE

but note that this has one independent variable (x) and one dependent variable (y)


$\displaystyle \frac{{dy}}{{dx}} = {x^2}$ is also a first order first degree ODE



$\displaystyle \frac{{\partial y}}{{\partial x}} = k\frac{{\partial y}}{{\partial t}}$

is a first order first degree PDE
Note that there are now two independent variables (x and t) and y is the dependent variable.

$\displaystyle \frac{{{d^2}y}}{{d{x^2}}} = A\cos x$ is a second order first degree ODE



$\displaystyle \frac{{{\partial ^2}\Omega }}{{\partial {v^2}}} = {\rm{Constant}}$

is a second order first degree PDE

However if you look carefully this is 'missing' one independent variable.
This can only happen if there is a second equation in which the second independent variable derivative is zero


$\displaystyle \frac{{{\partial ^2}\Omega }}{{\partial {u^2}}} = 0$


$\displaystyle {\left( {\frac{{dy}}{{dx}}} \right)^2} = x$

is a first order second degree ODE.


The solution of the first equation (an ODE)


$\displaystyle \frac{{dy}}{{dx}} = x$

is


$\displaystyle {\rm{y = }}\frac{1}{2}{x^2} + C$

Where C is an arbitrary constant.

And here is the most important difference from PDEs

You have to integrate the equation as many times as the order of the equation to produce a solution.

Each integration of an ODE introduces an arbitrary constant, so a first order has one constant, a second order two arbitrary constants and so on.

But for a PDE, each integration introduces an arbitrary function rather than a constant.

For both types of equation (additional) boundary conditions are required to determine the arbitrary constants or functions.
Normally, one condition per constant or function is required.
Thanks from awholenumber

Last edited by skipjack; July 18th, 2017 at 02:28 PM.
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July 18th, 2017, 12:57 PM   #14
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Each integration of an ODE introduces an arbitrary constant, so a first order has one constant, a second order two arbitrary constants and so on.

But for a PDE, each integration introduces an arbitrary function rather than a constant.
This is definitely a useful statement, but sadly not generally true. It's one of those things in math that should be generally true, but where annoying exceptions arise.

I highly recommend Bleecker's Basic partial differential equations to the OP; it actually discusses this in quite some detail.

Last edited by skipjack; July 18th, 2017 at 02:31 PM.
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July 18th, 2017, 01:42 PM   #15
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Quote:
Originally Posted by Micrm@ss View Post
This is definitely a useful statement, but sadly not generally true. It's one of those things in math that should be generally true, but where annoying exceptions arise.

I highly recommend Bleecker's Basic partial differential equations to the OP; it actually discusses this in quite some detail.
I think someone starting out in PDEs needs the broad brush before the pathogenic exceptions.

Heaven knows there are few enough physically important PDEs we can solve analytically anyway.
For most, we have to resort to numerical methods.

I don't know Bleecker, but thanks for the reference.
Thanks from awholenumber

Last edited by skipjack; July 18th, 2017 at 02:30 PM.
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August 17th, 2017, 03:27 AM   #16
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Sorry for the late response; I almost forgot about this thread.

Thanks a lot for explaining a lot of basic things studiot.

I have few more doubts.

Someone said many physics equations involve Ordinary and Partial differential equations.

The only two physics equations worth learning to me looks like Maxwell equations and Schrödinger equation.

Why I am a bit curious about these is because I see it in every physics documentary I see. And I would like to know more about those two equations.


I was trying to make a few rough notes, and I don't know whether these things I wrote down and right.


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Derivative is the rate of change of the dependent variable y with respect to the independent variable x, which is a number.

Differential equations are equations which involve one or more derivatives or the rate of change of the dependent variable y with respect to the independent variable x, or dy/dx, which is a number ...

Which is a number in an unknown function

Partial derivative

The character ∂ is a stylized d mainly used as a mathematical symbol to denote a partial derivative such as ∂z/∂x (read as "the partial derivative of z with respect to x").

A derivative of a function of two or more variables with respect to one variable, the other(s) being treated as a constant.

Partial differential equation

An equation containing one or more partial derivatives.


The character ∂ is a stylized d mainly used as a mathematical symbol to denote a partial derivative such as ∂z/∂x (read as "the partial derivative of z with respect to x").

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Ordinary differential equations for ex dy/dx tells something about the relation between dependent variable y to the independent variable x ... which is a number of a certain variables in an unknown function

Partial differential equations for ex SquigglyDy/dx tells something about the relation between dependent variable y to the independent variable x ... which is a number of a certain variables in an unknown function
What do the variables in those equations represent?

Last edited by skipjack; August 25th, 2017 at 08:37 AM.
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August 17th, 2017, 05:56 AM   #17
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None of those is an example of a partial differential equation.
Really
Then what are they ?
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August 17th, 2017, 07:01 AM   #18
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Quote:
Originally Posted by MATHEMATICIAN View Post
Really
Then what are they ?
They are ordinary differential equations.
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August 17th, 2017, 07:44 AM   #19
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They are ordinary differential equations.
I realize it just now
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August 25th, 2017, 08:05 AM   #20
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I have a few more questions to ask.

I have been writing down a few notes and I am not sure whether this is right.

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If I am thinking it right, on one side you have an unknown function,
on the other side you have a gradient, rate of change, dy/dx or measurement of the change of one dependent variable with respect to another independent variable.

Find the unknown function that caused that measurement of change ?

For that, you have to integrate that dy/dx or the gradient or the rate of change or the measurement of change of one dependent variable with respect to another independent variable.

And thus you reach a conclusion about the function that created that measurement of change right there.

But then what's the use of numerical methods?

Sometimes you can't find the exact function that created that measurement of change of one dependent variable with respect to another independent variable.

But with numerical methods you can approximate that function with numerical methods .

Quote:
For physicists the variables are whatever is being considered, such as time, length, energy, etc.
I was wondering what measurements of the variables are we taking into account in Equations like Maxwell's Equations and Schrödinger Equations.

In those equations, the variables could be about some measurements of electron, electricity, magnetism etc.


Are all those things I wrote down right?

Last edited by skipjack; August 25th, 2017 at 08:20 AM.
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