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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 tex2html_wrap_inline231 . 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, tex2html_wrap_inline243 is the partial derivative of B(x,t) with respect to x while holding t constant. Similarly tex2html_wrap_inline251 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 tex2html_wrap_inline259 is the second derivative of p(x,t) with respect to t, holding x constant, and tex2html_wrap_inline267 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