» Required Reading » Table of Contents » Chapter 6 6.5 The Tangent Plane and the Gradient Vector
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We define differentiability in two dimensions as follows. A function f of two variables is differentiable at argument (x₀, y₀) if the surface it defines in (x, y, f) space looks like a plane for arguments near (x₀, y₀). (Given any positive numerical criterion, there is a circle around (x₀, y₀) within which its graph differs from the plane by less than that criterion). A plane in variables f, x and y is defined by a linear equation that can be put in the form

f(x, y) = a (x - x₀) + b (y - y₀) + f(x₀, y₀).     (A)

The plane that f resembles here is called the tangent plane to f at (x₀, y₀) and the function it represents is called the linear approximation to f at that point. The quantities a and b are called the partial derivatives of f with respect to x and with respect to y, and written as follows:

Here a is the directional derivative of f in the direction of the x axis, i, while b is the directional derivative of f in the direction of the y axis, j.

The linear approximation to f(x, y) at arguments x₀ and y₀ which describes the tangent plane to f at (x₀, y₀) therefore takes the form:

where the vector (grad f), called the gradient vector to f at (x₀, y₀), is the vector whose components are the partial derivatives of f in the x and y directions at the point (x₀, y₀).

We generally do not write out the cumbersome subscripts that indicate the point (x₀, y₀) at which the gradient and linear approximation are defined, because they are so cumbersome, and simply write

Notice that we can write grad f as with the symbol , called "del" representing the combination:.