partial derivatives, gradient, divergence, and curl partial ......partial derivatives recall that...

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Partial Derivatives, Gradient, Divergence, and Curl

Partial Derivatives

Recall that the derivative of a function of one variable provides information about how the function changes as the input variable changes. For a function of multiple variables, we consider how the function changes when one input variable changes and all other input variables are fixed, and refer to partial derivatives. For example, if is a function of two variables and , then its partial derivatives are the functions and , which are defined by

provided that the limit(s) exist.

Notation for Partial Derivatives

For a function , all of the following notation is used to represent partial derivatives.

Rule for Finding Partial Derivatives of

1. To find , regard as a constant and differentiate with respect to . 2. To find , regard as a constant and differentiate with respect to .

fx y xf yf

( ) ( ) ( ) ( ) ( ) ( ) 0 0

, , , , , lim and , lim x yh h

f x h y f x y f x y h f x yf x y f x y

h h® ®

+ - + -= =

( ), z f x y=

( ) ( ) 1 1, , x x xf zf x y f f x y f D f D fx x x¶ ¶ ¶

= = = = = = =¶ ¶ ¶

( ) ( ) 2 2, , y y yf zf x y f f x y f D f D fy y y¶ ¶ ¶

= = = = = = =¶ ¶ ¶

( ), z f x y=

xf y ( ), f x y x

yf x ( ), f x y y


Partial derivatives are similarly defined for functions of three or more variables.

Higher Order Derivatives

For higher order derivatives, we take derivatives one order at a time. For a function , we use the notation

The notation is similar for higher order partial derivatives.

Clairaut’s Theorem

Suppose that is defined on a disk that contains the point . If the functions

and are both continuous on , then . In essence, if is a “nice” function, then a mixed partial derivative is independent of the order we take the derivatives, i.e. .

Del Operator

The del operator is denoted as which is called nabla. The del operator is a vector written as

Basically, the presence of the tells us we are taking a derivate, which takes several forms, each of which gives different results!

- If is a function, then (gradient) is a vector.

( ), z f x y=

2 2

11 2 2( )x x xxf f zf f f

x x x x¶ ¶ ¶ ¶æ ö= = = = =ç ÷¶ ¶ ¶ ¶è ø

2 2

12( )x y xyf f zf f f

y x y x y x¶ ¶ ¶ ¶æ ö= = = = =ç ÷¶ ¶ ¶ ¶ ¶ ¶è ø

2 2

21( )y x yxf f zf f f

x y x y x yæ ö¶ ¶ ¶ ¶

= = = = =ç ÷¶ ¶ ¶ ¶ ¶ ¶è ø

2 2

22 2 2( )y y yyf f zf f f

y y y yæ ö¶ ¶ ¶ ¶

= = = = =ç ÷¶ ¶ ¶ ¶è ø

f D ( ), a b

xyf yxf D ( ) ( ), , xy yxf a b f a b= f

xy yxf f=

Ñ

i j kx y z

æ ö¶ ¶ ¶æ ö æ öÑ = + +ç ÷ç ÷ ç ÷¶ ¶ ¶è ø è øè ø

Ñ

f fÑ


- If is a vector field, then (divergence) is a scalar function. - If is a vector field, then (curl) is a vector function.

We will consider each of these forms below.

Gradient

If is a function of two variables and , then the gradient of is the vector defined by

where and are unit vectors.

The gradient of a function is similarly defined for functions of three or more variables.

We think of the gradient of a function of two or more variables as the direction of the fastest increase of at a point, with magnitude equal to the maximum rate of increase at a point – just as we do for the derivative of a single-variable function. Geometrically, the gradient at a point is perpendicular to the tangent vector of any curve that passes through the point.

Example. If , find the gradient at the point .

Solution. Taking the required partial derivatives, we obtain

Plugging in the point , we obtain

Divergence

If , where , , and are functions, is a vector field on , and ,

, and exist, then the divergence of is defined by

F FÑ×F FÑ´

f x y f fÑ

( ) ( ) ( ) ( )grad , , , , , x yf ff x y f x y f x y f x y i jx y¶ ¶

=Ñ = = +¶ ¶

i j

f

( ) 2 3, 2 4 5f x y x y y= - + ( )2, 1-

( ) ( ) ( ) 3 2 2, , , , 4 , 6 4 .x yf x y f x y f x y xy x yÑ = = -

( )2, 1-

( ) ( )( ) ( ) ( )3 2 22, 1 4 2 1 ,6 2 1 4 8, 20 .fÑ - = - - - = -

F Pi Qj Rk= + + P Q R 3R Px¶¶

Qy

¶¶

Rz¶¶

F

div P Q RF Fx y z¶ ¶ ¶

Ñ× = = + +¶ ¶ ¶


The divergence of a function is similarly defined for functions of an arbitrary number of variables.

In essence, the divergence is the dot product of the del operator and the vector field which results in a scalar function.

Example. If , then find .

Solution. We note that . Applying the definition, we obtain

Curl

If , where , , and are functions, is a vector field on , and ,

, and exist, then the curl of is defined by

Or equivalently, .

The curl of a function is not defined for functions with a different number of variables; it only applies to 3-dimensional space.

In essence, the curl is the cross product of the del operator and the vector field which results in a vector field.

Geometrically, the curl of is a vector that is perpendicular to the vector field that represents the velocity of a particle in motion. Its length and direction characterize the rotation at a particular point.

Example. If , then find .

Solution. Applying the definition, we obtain

ÑF

F xzi xyzj= + FÑ×

0R =

( ) ( ) ( )0 .P Q RF xz xyz z xzx y z x y z¶ ¶ ¶ ¶ ¶ ¶

Ñ× = + + = + + = +¶ ¶ ¶ ¶ ¶ ¶

F Pi Qj Rk= + + P Q R 3R Px¶¶

Qy

¶¶

Rz¶¶

F

curl R Q P R Q PF F i j ky z z x x y

æ ö æ ö¶ ¶ ¶ ¶ ¶ ¶æ öÑ´ = = - + - + -ç ÷ ç ÷ç ÷¶ ¶ ¶ ¶ ¶ ¶è øè ø è ø

i j k

Fx y zP Q R

¶ ¶ ¶Ñ´ =

¶ ¶ ¶

Ñ F

F

2 2 2 3F xyz i x yj x yz k= - + FÑ´


Fun Facts:

Theorem. If is a function of three variables that has continuous second-order partial derivatives, then

It follows that if is conservative, meaning there is a function such that , then

Theorem. If is a vector field on that has continuous second-order partial derivatives, then

Proof.

( )

( ) ( ) ( )

2 2 2 3

2 3 2 2 2 3 2 2

2 3 3 2

( )

2 2 2 .

i j k

Fx y zP Q R

i j k

x y zxyz x y x yz

x yz x y i xyz x yz j x y xyz ky z z x x y

x z i xyz xyz j xy xz k

¶ ¶ ¶Ñ´ =

¶ ¶ ¶

¶ ¶ ¶=

¶ ¶ ¶-

æ ö æ ö¶ ¶ ¶ ¶ ¶ ¶æ ö= - - + - + - -ç ÷ ç ÷ç ÷¶ ¶ ¶ ¶ ¶ ¶è øè ø è ø

= + - + - -

f

( )curl 0.f fÑ´Ñ = Ñ =

F f F f=Ñ

( )curl 0.F FÑ´ = =

F Pi Qj Rk= + + 3R

( ) div curl 0F FÑ× Ñ´ = =


since mixed partial derivatives do not depend on the order of the variables.

( )

2 2 2 2 2 2

0,

i j k

Fx y zP Q R

R Q P R Q Pi j ky z z x x y

R Q P R Q Px y z y z x z x y

R Q P R Q Px y x z y z y z z x z x

¶ ¶ ¶Ñ × Ñ´ =Ñ×

¶ ¶ ¶

æ öæ ö æ ö¶ ¶ ¶ ¶ ¶ ¶æ ö=Ñ× - + - + -ç ÷ç ÷ ç ÷ç ÷¶ ¶ ¶ ¶ ¶ ¶è øè ø è øè øæ ö æ ö¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶æ ö= - + - + -ç ÷ ç ÷ç ÷¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶è øè ø è ø

¶ ¶ ¶ ¶ ¶ ¶= - + - + - =¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶

partial derivatives, gradient, divergence, and curl partial ......partial derivatives recall that...

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