Differentiation

The Root of All Calculus


Calculus or mathematical analysis is built up from 2 basic ingrediants: integration and differentiation. Differentiation is concerned with things like speeds and accelerations, slopes and curves ect. These are Rates of Change, they are things that are defined locally. The Fundamental Theorem of Calculus is that Integration and Differentiation are the inverse of each other.

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Separation of Variables


Wave equations can be difficult to solve as they can have more than one solution and can look hideously difficult. One way of solving partial differential wave equations is to use separation of variables.

To seperate the variables you have to assume that the wave equation you are trying to solve can be split up into two different functions, one of x and one of t. So you assume your equation can be written in this form,

Where X is a function of distance (x) only, and T is a function of time (t) only. So lets try this method on a wave equation,

So if we take the left hand side first we need to differentiate our assumed equation with respect to x twice, which is

The T sticks around as it is a constant with respect to x. For the right hand we get

Once again, the X doesn't disappear as it is just classed as a constant. Now we can equate these two things like they are in the wave equation

Rearranging them gives

The reason we rearranged them was to get all the X's on one side and all the T's on the other. This shows that a function of x on the left equals something that has nothing to do with x on the right. This can only happen if both sides of the equation equal a constant. You can choose whatever symbol you want for this constant but unless you pick it as negative you'll have imaginary numbers and complex numbers to deal with. So you get

and

I have taken the X and T to the other side with the constant so now we have two relatively simple differential equations.

For X we have something that when differentiated twice with respectt to x gives us the original thing plus a factor of -α2, and for T we have something that when differentiated twice with respect to t gives us the original plus a factor of -α2c2. The only way we can get these is with sin or cos functions, so the options are

Without any boundary conditions there is no way to get rid of any of the answers, so the solutions are

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Div, Grad and Curl


When you need to find the Divergence, Gradient or Curl of a vector field or scalar field you basically need to know one main operator. This operator is called Del, and looks like this

You get the div gad or curl depending on how you use del. If for example you just use it to operate on a scalar field T the you get

Del operating on a scalar give a vector answer corresponding to the divergence of the field. If however you decide to take the dot product of del with a vector field you would get a scalar answer, like so

Taking the cross product of del with a vector field gives a vector answer that corresponds to the curl of the field

You can also use del multiple times. In the below example the gradient of the scalar field T was calculated, then dotted with del to calculate the divergence of the gradient

This is usually reffered to as del-squared or the Lapalce operator and is equal to


  •The Root of
   all Calculus

  •Separation
   of Variables

  •Div, Grad
   and Curl



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