# I What are partial differential equations?

1. Jun 15, 2017

### awholenumber

If the slope of the curve (derivative) at a given point is a number .

2. Jun 15, 2017

### BvU

3. Jun 15, 2017

### awholenumber

What are partial differential equations ?

4. Jun 15, 2017

### BvU

Do you want PF to play textbook for you ? What did you find so far and what is unclear ?

5. Jun 15, 2017

### awholenumber

I was wondering if anybody could explain this in this context

6. Jun 15, 2017

### Staff: Mentor

This question has no relation to the title. Partial differentia equations are not the same thing as derivatives (even though they are related).

7. Jun 15, 2017

### awholenumber

I was hoping somebody would explain the relationship between the two

8. Jun 15, 2017

### BvU

Where are you in your curriculum ? You seem to know what a derivative is -- for a function of a single independent variable, e.g. $f(x)$. Right ?

Partial derivatives are a kind of extension to functions of several variables such as $f(x,y)$: if you keep y constant, e.g. at $Y_1$ then the partial derivative wrt x of $f$ is the derivative of the now single variable function $f(x, Y_1)$.

But you have googled such things yourself already, so I ask again: what did you find so far and what is unclear ?

(and perhaps also: what textbook are you using ?)

9. Jun 15, 2017

### awholenumber

I was going through this website here ,
https://17calculus.com/differential-equations/

Is it possible to understand partial differential equations in terms of a working example of something ?

This is the only thing i could think of .

10. Jun 15, 2017

### BvU

Wrong website. This is about differential equations. Before going through that you should become familiar with differentials. An introductory calulus textbook or website if the Wiki link I gave you is too abstract.

Sure. Once you're comfortable wit Algebra, Calculus I and II, there are plenty in Calculus III on partial derivatives

I'm not trying to discourage you: it's more a matter of finding out what foundations you have as a start for your adventure

Not clear what you aim at with
I suspect you have some specific topic that interests you ?

11. Jun 15, 2017

### awholenumber

It is ok , i sort of understand it with my new avatar .
Some points i tried to narrow down .

Differential calculus - which is looking at the instantaneous rates of change of objects with respect to some variables. We have the notion of the derivative of a function
The slope of the curve (derivative) at a given point is a number

Partial derivative - A derivative of a function of two or more variables with respect to one variable, the other(s) being treated as constant.
An equation that involves the derivatives of a function of several variables is a "partial differential equation"

12. Jun 15, 2017

### Staff: Mentor

All you're doing here is giving definitions of a few terms, a very long way from being able to solve ordinary differential equations (involving single-variable functions) or partial differential equations (involving multi-variable functions).

BTW, what does having a new avatar have to do with understanding anything?

13. Jun 15, 2017

### awholenumber

Thanks ,

First you have a function f(x)
Then you can take the derivative of that function from first principles
Then you get f'(x) or dy/dx

That dy/dx , the slope of the curve (derivative) at a given point is a number .

I have been unnecessarily thinking about the cause behind why differential equations have dy/dx or f'(x) in them .I guess there is no need to think about it that way .

14. Jun 15, 2017

### WWGD

In a sense, these are equations where the " input data" is given in terms of partial derivatives: you are given a numerical relation involving an unknown function and some of its partial derivatives and your goal is to identify the function(s) satisfying said relations, e.g., you are given that

$a(x) \frac {\partial^2 f}{\partial x \partial y} + b(x)f=c(x); a(x), b(x), c(x)$ are functions, and you need to figure out which functions f satisfy the equation. One you have likely seen is the Laplacian of a function $U=U(x,y,...): U_{xx}+ U_{yy}+...=0$ for a function U of many variables, or the Cauchy-Riemann equations , :https//en.wikipedia.org/wiki/Laplace_operator and https://en.wikipedia.org/wiki/Cauchy–Riemann_equations respectively

15. Jun 16, 2017

### awholenumber

Thanks :-)

16. Jun 16, 2017

### Staff: Mentor

Differential equations are equations that involve the derivatives of various orders of some unknown function f. These derivatives represent the rates of change of some quantity with respect to some other quantity. You don't start off knowing the function, so it makes no sense to talk about finding its derivative.

For example, a mass on a spring, with no damping force, can be represented by the differential equation $m \frac{d^2 }{dt^2}\left(x(t)\right) + kx(t) = 0$ or $mx''(t) + kx(t) = 0$. Solving this equation involves finding the position x(t). See https://en.wikipedia.org/wiki/Harmonic_oscillator#Spring.2Fmass_system

17. Jun 16, 2017

### awholenumber

Thanks :-)

18. Jun 17, 2017

### awholenumber

Which one is the independent variable and the dependent variable here?

19. Jun 17, 2017

### Staff: Mentor

Look at what you quoted in post #9.

I don't know why you are asking questions about partial differential equations if you can't look at an equation and tell which variable is dependent and which is independent.

20. Jun 17, 2017

### awholenumber

Ok , The independent variable is x. The dependent variable is f(x)

In this ,

The function 1 , 4 , 5 are functions of several variables right ?