6.5 The Tangent Plane and the Gradient Vector

We define differentiability in two dimensions as follows. A function $f$ of two variables is differentiable at argument $(x_{0}, y_{0})$ if the surface it defines in $(x, y, f)$ space looks like a plane for arguments near $(x_{0}, y_{0})$ .

(Given any positive numerical criterion, there is a circle around

(x_{0}, y_{0})

within which its graph differs from the plane by less than that criterion.)

Recall that a plane in variables

f, x

and

y

is defined by a linear equation that can be put in the form

The plane that

f

resembles here is called the tangent plane to $f$ at $(x_{0}, y_{0})$ and the function it represents is called the linear approximation to $f$ defined at $(x_{0}, y_{0})$ .

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

a = \frac{\partial f}{\partial x}; b = \frac{\partial f}{\partial y}

Here

a

is the directional derivative of

f

in the direction of the

x

axis,

\hat{i}

, while

b

is the directional derivative of

f

in the direction of the

y

axis,

\hat{j}

The linear approximation to

f (x, y)

at arguments

x_{0}

and

y_{0}

which describes the tangent plane to

f

(x_{0}, y_{0})

therefore takes the form

f L (x, y) = {\frac{\partial f}{\partial x} |}_{(x_{0}, y_{0})} (x - x_{0}) + {\frac{\partial f}{\partial y} |}_{(x_{0}, y_{0})} (y - y_{0}) + f (x_{0}, y_{0}) = f (x_{0}, y_{0}) + (\overset{⟶}{grad} f) \cdot (\overset{⟶}{r} - \overset{⟶}{r_{0}})

where the vector

\overset{⟶}{grad} f

, called the gradient vector to $f$ at $(x_{0}, y_{0})$ , is the vector whose components are the partial derivatives of

f

in the

x

and

y

directions at the point

(x_{0}, y_{0})

\overset{⟶}{grad} {f |}_{(x_{0}, y_{0})} = ({\frac{\partial f}{\partial x} |}_{(x_{0}, y_{0})}, {\frac{\partial f}{\partial y} |}_{(x_{0}, y_{0})})

We generally do not write out the cumbersome subscripts that indicate the point

(x_{0}, y_{0})

at which the gradient and linear approximation are defined, because they are so cumbersome, and simply write

\overset{⟶}{grad} f = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}) = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j}

Notice that we can write

\overset{⟶}{grad} f

(\frac{\partial}{\partial x} \hat{i} + \frac{\partial}{\partial y} \hat{j}) f

\overset{⟶}{\nabla} f

with the symbol

\overset{⟶}{\nabla}

, called "del" representing the combination:

(\frac{\partial}{\partial x} \hat{i} + \frac{\partial}{\partial y} \hat{j})

Home \| 18.013A \| Chapter 6		Tools Glossary Index Up Previous Next