If you took a look at one of the previous posts on how to remember the Laplacian in different forms by using a metric, you will notice that the form of the Laplacian that we get is:
But in Jackson’s Classical Electrodynamics, III edition he notes the following:
This is an interesting form of the Laplacian that perhaps not everyone has encountered. This can obtained from the known form by making the substitution and simplifying. The steps to which have been outlined below:
To understand why this is true, we must start with the Fundamental Theorem of Vector calculus. If is a conservative field ( i.e ), then
What this means is that the value is dependent only on the initial and final positions. The path that you take to get from A to B is not important.
Now if the path of integration is a closed loop, then points A and B are the same, and therefore:
Now that we are clear about this, according to Stokes theorem the same integral for a closed region can be represented in another form:
From this we get that Curl = for a conservative field (i.e ). Therefore when a conservative field is operated on by a curl operator (), it yields 0.
Bravo Prof.Ghrist! Beautifully said 😀
In this post, let’s derive a general solution for the Laplacian in Spherical Coordinates. In future posts, we shall look at the application of this equation in the context of Fluids and Quantum Mechanics.
The Laplacian in Spherical coordinates in its ultimate glory is written as follows:
To solve it we use the method of separation of variables.
Plugging in the value of into the Laplacian, we get that :
Dividing throughout by and multiplying throughout by , further simplifies into:
It can be observed that the first expression in the differential equation is merely a function of and the remaining a function of and only. Therefore, we equate the first expression to be and the second to be . The reason for choosing the peculiar value of is explained in another post.
The first expression in (1) the Euler-Cauchy equation in .
The general solution of this has been in discussed in a previous post and it can be written as:
The second expression in (1) takes the form as follows:
The following observation can be made similar to the previous analysis
The first expression in the above equation (2) is the Associated Legendre Differential equation.
The general solution to this differential equation can be given as:
The solution to the second term in the equation (2) is a trivial one:
Therefore the general solution to the Laplacian in Spherical coordinates is given by: