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Review of partial derivatives

Since you probably don't use partial derivatives everyday and may feel a bit rusty with them, it is probably good for us to review their use.

If B is a function of x, we denote it as B(x) and its derivative (slope) at x by . It is clear what is going on because B is a function of a single variable.

However if B is a function of more than one variable, say both x and t, then then we denote it as B(x,t) and we can take derivatives (slopes) in more than one way. The easiest way to make things clear is to use a partial derivative where we take the derivative with respect to one variable while holding the other variable constant.

Hence, is the partial derivative of B(x,t) with respect to x while holding t constant. Similarly is the partial derivative of B(x,t) with respect to t while holding x constant.

As you can see on the previous page, the wave equation involves taking two partial derivatives each for both variables of interest. For example is the second derivative of p(x,t) with respect to t, holding x constant, and is the second derivative of p(x,t) with respect to x, holding t constant.

VICTOR W SPARROW
Tue Feb 25 21:24:33 EST 1997