What's with this separation of variables business?

In summary, Frank is feeling overwhelmed by electromagnetism and separation of variables. He doesn't understand the derivation up to the point where he has X\frac{d^2 X}{d x ^2}=C_1 , \quad Y\frac{d^2 Y}{d y ^2}=C_2, \quad Z\frac{d^2 Z}{d z ^2}=C_3 with C_1+C_2+C_3=0. He is wondering if there is a better way to understand SOV than through math.
  • #1
Ai52487963
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Electromagnetism just got weird. REALLY weird. Everything was going great until we hit this new chapter on separation of variables. I don't remember doing this kind of stuff in my DiffEqs class.

Frankly, I'm feeling overwhelmed. I have a midterm at the end of this week, and I feel as though if I were to be tested on boundary conditions of electric potentials, then I'm doomed for sure. Multipole expansions make more sense to me and seem far, far less hand-wavey than separation of variables does.

Is there a better way of understanding separation of variables than all this wacky math stuff? This n and m business combined with double summations is getting out of hand. Fast.
 
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  • #2
I don't think separation of variables are handwaveying, maybe it is just your teacher who gives you this impression? Separation of variables is a great way to solve Partial Differential equations.

We did have sep. of var. both in our transform methods - math class and in mathematical methods of physics class.
 
  • #3
Perhaps you can be more specific about which aspect(s) of SOV you find hand-wavy?
 
  • #4
gabbagabbahey said:
Perhaps you can be more specific about which aspect(s) of SOV you find hand-wavy?

SOV made sense to me in my differential eqns class perfectly, but in my book, Introduction to Electrodynamics by Griffiths, he goes over an example of an infinitely long rectangular pipe with three sides grounded and one end of the pipe at V.

He goes through boundary conditions and applies Laplace's equation and arrives at some double summation for V(x,y,z) involving nearly a million arbitrary constants. I guess its the part of SOV involving the part where he takes his X(x), Y(y) and Z(z) equations and applies the boundary conditions to them is where I get lost.
 
  • #5
Are you referring to Example 3.5 in the 3rd edition?

Are you okay with the derivation up to the point where he has

[tex]X\frac{d^2 X}{d x ^2}=C_1 , \quad Y\frac{d^2 Y}{d y ^2}=C_2, \quad Z\frac{d^2 Z}{d z ^2}=C_3 [/tex] with [tex]C_1+C_2+C_3=0[/tex]

Or is there anything up until thet point that you don't understand?
 

1. What is separation of variables?

Separation of variables is a method used in solving differential equations. It involves breaking down a complex equation into simpler parts that can be solved individually. This approach is often used when an equation has multiple variables that are not directly related to each other.

2. Why is separation of variables used?

Separation of variables is used to simplify the process of solving differential equations. It allows us to solve each part of the equation separately, making the overall solution easier to obtain. It also works well for equations with boundary conditions and is applicable to a wide range of problems in physics and engineering.

3. What are the steps involved in separation of variables?

The first step is to identify the variables in the equation and separate them on opposite sides. Then, we assume a solution in the form of a product of functions, each depending on only one variable. Next, we substitute this solution into the original equation and simplify. Finally, we solve for each individual function by using appropriate boundary conditions.

4. Are there any limitations to using separation of variables?

Yes, there are limitations to using separation of variables. This method is only applicable to linear partial differential equations with constant coefficients. It also assumes that the variables can be separated and that the solution can be expressed as a product of functions. In some cases, these assumptions may not hold, and other methods may need to be used.

5. Can separation of variables be used for all types of differential equations?

No, separation of variables can only be used for certain types of differential equations, such as linear partial differential equations with constant coefficients. It is not applicable to nonlinear equations or equations with variable coefficients. In these cases, other techniques, such as numerical methods or series solutions, may be used.

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