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Suppose we have a function of two variables, .
Such things are sometimes called scalar fields . (Scalar to indicate they are not vectors, and fields to indicate that there are two or more variables.)
We can choose a particular line in plane, (for example ) and consider the function of (with everything else in it fixed): .
is then called
the directional derivative of
at
in the direction in the
plane having slope
.
In other words we can, by picking out any particular line in the plane, reduce to a function of a single value defined on that line and define the derivative of that one variable function with respect to distance on that line.
This derivative is called the directional derivative of in the direction of the line. (You may examine the directional derivatives of functions of two variables in the applet .)
The directional derivative in the direction of the x-axis is called the partial derivative of with respect to , and is written as .
Similarly the directional derivative of in the direction of the y-axis is called the partial derivative of with respect to , and is written as .
These partial derivatives are computable exactly as ordinary one dimensional derivatives are. When computing the partial derivative with respect to , you treat as a constant, and differentiate with respect to exactly as you do in one dimension.
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