]> 6.5 The Tangent Plane and the Gradient Vector

## 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

$f ( x , y ) = a ( x − x 0 ) + b ( y − y 0 ) + f ( x 0 , y 0 )$      (A)

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

$a = ∂ f ∂ x ; b = ∂ f ∂ y$

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 $x 0$ and $y 0$ which describes the tangent plane to $f$ at $( x 0 , y 0 )$ therefore takes the form

$f L ( x , y ) = ∂ f ∂ x | ( x 0 , y 0 ) ( x − x 0 ) + ∂ f ∂ y | ( x 0 , y 0 ) ( y − y 0 ) + f ( x 0 , y 0 ) = f ( x 0 , y 0 ) + ( grad ⟶ f ) · ( r ⟶ − r 0 ⟶ )$

where the vector $grad ⟶ f$ , 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 $( x 0 , y 0 )$

$grad ⟶ f | ( x 0 , y 0 ) = ( ∂ f ∂ x | ( x 0 , y 0 ) , ∂ f ∂ y | ( x 0 , y 0 ) )$

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

$grad ⟶ f = ( ∂ f ∂ x , ∂ f ∂ y ) = ∂ f ∂ x i ^ + ∂ f ∂ y j ^$

Notice that we can write $grad ⟶ f$ as $( ∂ ∂ x i ^ + ∂ ∂ y j ^ ) f$ or $∇ ⟶ f$ with the symbol $∇ ⟶$ , called "del" representing the combination: $( ∂ ∂ x i ^ + ∂ ∂ y j ^ )$ .