Chap. 9 Vector Differential Calculus. Grad, Div, Curl

중앙대학교 건설환경플랜트공학과 교수

김 진 홍

- 8주차 강의 내용 -

9.6 Calculus Review: Functions of Several Variables.

Chain Rules Theorem 1

Let be continuous and be

functions that are continuous. Then the function is defined and

(1)

) , , (x y z f

w x x(u,v), y  y(u,v), z  z(u,v) ))

, ( ), , ( ), , (

(x u v y u v z u v f

w

v z z w v y y w v x x w v

w

u z z w u y y w u x x w u w







 







 







 











 







 







 





Special Cases of Practical Interest

- If w = f(x, y) and x = x(u, v), y = y(u, v), then (1) becomes

(2)

v y y w v

x x w v

w

u y y w u

x x w u

w







 







 











 







 





Chap. 9 Vector Differential Calculus. Grad, Div, Curl

If w = f(x, y, z) and x = x(t), y = y(t), z = z(t) then (1) gives

(3) dt

dz z w dt dy y w dt dx x w dt dw













If w = f(x, y) and x = x(t), y = y(t), then (3) reduces to (4) ^dw_dt ^{ }_^w_x ^dx_dt ^_^w_y ^dy_dt

Finally, the simplest case w = f(x), x = x(t) gives

(5) dt

dx dx dw dw dt

Ex. 1) w x² y² _and xrcos



, yrsin



, then









 2 sin 2 cos 2 sin 2 cos 2 cos

2x y r ² r ² r

r

w     

















 ^{2 ( sin}^{) 2 ( cos ) 2 cos sin 2 sin cos 2 sin2}

2 2

2 r r

r r

y r

w x     





Ex. 2) z e^xsin y, x st², y s²t

) cos(

2 ) sin(

2 ) cos ( ) sin

(e y t² e y st t²e ² s²t ste ² s²t s

y y z s x x z s

z  x  x  st  st





















) cos(

) sin(

2 ) cos ( 2 ) sin

(e y st e y s² ste ² s²t s²e ² s²t t

y y z t x x z t

z _{x x st st}







 



















Chap. 9 Vector Differential Calculus. Grad, Div, Curl

Mean Value Theorem Theorem 2

Let be continuous, and and be such that the straight line segment

P

₀

P

. Then

(6)

the partial derivatives being evaluated at a suitable point of that segment.

) , , (x y z f

w ^P₀^:(x₀^,y₀^,z₀⁾ P^:(x0h^,y0k^,z0l⁾

z l f y k f x h f z y x f l z k y h x

f 

 



 



 







 , , ) ( , , )

( _{0 0 0 0 0 0}

Special Cases

- For a function f(x, y) of two variables, (6) reduces to

(7) y

k f x h f y x f k y h x

f 

 



 





 , ) ( , )

( _{0 0 0 0}

and for a function f(x) of a single variable, (6) becomes

(8) dx

hdf x

f h x

f( ₀ ) ( ₀)

Chap. 9 Vector Differential Calculus. Grad, Div, Curl

Mean value theorem for a function of two variables

9.7 Gradient of a Scalar Field. Directional Derivative.

Definition Gradient

The gradient of a given scalar function f(x, y, z) is denoted by grad f or ∇f and is the vector function defined by

(1)

Here x, y, z are Cartesian coordinates in a domain in 3-space in which f is defined and differentiable.

z k j f y i f x f z f y f x f f



 



 



 











 

 [ , , ]

- The differential operator ∇ is defined by

(1*) k

j z i y

x 

 



 



 



x xz y

z y x

f( , , )2 ³4 3

*

] 4 , 6 , 3 4

[ z y² x

f

grad  

Chap. 9 Vector Differential Calculus. Grad, Div, Curl

The directional derivative or of a function f(x, y, z) in the direction of a vector b is defined by

(5)

If the direction is given by a vector a of any length (≠0), then (6)

Ex. 1) Find the directional derivative of at P:(2,1) in the direction of v = 2i+5j.

Sol) v  29

29 ) 32 40 8 29 ( ] 1

8 , 4 [ ] 5 , 2 29 [ ) 1

(P       

f D_a

y y x y x

f( , ) ^{2 3}4

Directional Derivative

f grad ds b

f df

D_b   

f grad a a

ds f df

D_a   1 

Chap. 9 Vector Differential Calculus. Grad, Div, Curl

ds f df D_b

) 1 ( b 

)], 4 3

( , 2

[ ^{3 2 2}

 xy x y f

grad gradf(P)[4,8]

Ex. 3) Find the directional derivative of at P:(1,3,0) in the direction of a = [1, 2, -1].

Sol) a  6

2 ) 6

3 6 ( ] 1

3 , 0 , 0 [ ] 1 , 2 , 1 6 [ ) 1

(P       

f D_a

The minus sign indicates that at

P

the function is decreasing in the direction of a.

yz x z y x

f( , , ) sin

Chap. 9 Vector Differential Calculus. Grad, Div, Curl

Ex. 2) Find the directional derivative of at P:(2,1,3) in the direction of a = [1, 0, -2].

Sol) grad

f =

[4x, 6y, 2z], grad

f

(P) = [8, 6, 6], a  5

789 . 1 ) 12 0 8 5 ( ] 1

6 , 6 , 8 [ ] 2 , 0 , 1 5 [ ) 1

(P        

f D_a

2 2

2 3

2 ) , ,

(x y z x y z

f   

], cos ,

cos ,

sin

[ yz xz yz xy yz f

grad  gradf(P)[0,0,3]

Gradient Is a Vector. Maximum Increase

Theorem 1

Vector Character of Gradient. Maximum Increase

Let

f(P) = f(x, y, z)

be a scalar function. Then grad

f

is a vector and its length and direction are independent of the Cartesian coordinates. If grad

f(P) ≠

0 at some point

P

, it has the direction of maximum increase of

f

at

P

.

Theorem 2

Gradient as Surface Normal Vector

Let

f

be a differentiable scalar function. Let

f(x, y, z)

= c represent a surface

S

. Then if the gradient of

f

at a point

P

of

S

is not zero vector, it is a normal vector of

S

at

P

.

Gradient as surface normal vector

Chap. 9 Vector Differential Calculus. Grad, Div, Curl

Ex. 2) Find a unit normal vector n of the cone of revolution

at a point ^{4( )}

2 2

2 x y

z  

).

2 , 0 , 1 ( : P

Sol.) ^{f(x, y, z)4(x2y2)z2}

grad f [8x,8y,2z], gradf(P)[8,0,4] 5]

, 1 0 5 , [ 2 ) ) (

(

1  

 grad f P

P f n grad

n points downward since it has a negative z-component.

Theorem 3

Laplace's Equation

The Laplace's equation is defined by

(9) 2

2 2 2 2 2 2

z f y

f x

f f











 



- ²f is also denoted by The differential operatorf. (10) ^{2 2}₂ ²₂ ²₂

z y

x 

 



 



 





 is called the Laplace operator.

Chap. 9 Vector Differential Calculus. Grad, Div, Curl

Cone and unit normal vector n

(1)

z v y v x v v

div 

 



 



  ^{1 2 3}

is called the divergence of v. For example,

v = [3xz,2xy,yz²]  3xzi2xyjyz²k, then div v3z2x2yz Another common notation for the divergence is

z v y

v x

v

k v j v i v z k

y j x i

v v z v

y v x

v div



 



 



 





 

 



 



 

 









 







3 2

1

3 2

1 3 2 1

] [

) (

] , , [ ] , , [

Note that ∇ v means the scalar div v, whereas ∇

f

means the vector grad

f

defined in Sec. 9.7.



9.8 Divergence of a Vector Field

Let v(x, y, z) be a differentiable vector function, where x, y, z are Cartesian coordinates, and let v₁ , v₂ , v₃ be the components of v.

Then the function

Chap. 9 Vector Differential Calculus. Grad, Div, Curl

Theorem 1

Invariance of the Divergence

The divergence

div v

is a scalar function, that is, its values depend only on the points in space (and, of course, on v) but not on the choice of the coordinates in (1), so that with respect to other Cartesian coordinates

and corresponding components of v,

(2) div v =

*

*

*, y , z

x v₁^*, v₂^*, v₃^*

*

* 3

*

* 2

*

* 1

z v y v x v













Let

f(x, y, z)

be a twice differentiable scalar function. Then its gradient exists,

2 2 2 2 2 2

) (

] , , [

z f y

f x

f f grad div v div

z k j f y i f x f z f y f x f f

grad v



 



 



 





 



 



 











 



Hence, the divergence of the gradient is the Laplacian.

(3) ^{div (grad f) 2f}

Chap. 9 Vector Differential Calculus. Grad, Div, Curl

Ex. 2) Flow of a Compressible Fluid. Physical Meaning of the Divergence

We consider the fluid motion in a region R having no sources, or sinks, and consider the flow through a rectangular box B.

k v j v i v v v

v₁, ₂, ₃] _{1 2 3}

[   



(4) uv [u₁,u₂,u₃]  u₁iu₂ju₃k and assume that u and v are continuously differentiable vector functions of x, y, z and t.

The mass of fluid entering through the left face ΔxΔz during a short time interval Δt is given by

where the subscript y indicates the left face.

t z x u t z x

v )_y   ( )_y  

( _{2 2}

The mass of fluid leaving the box B through the opposite face during the same time interval is approximately

where the subscript indicates the right face.

] ) ( ) (

[ _{2 2 2}

2

2 V t u u y y u y

y t u z x

u     











 _

t z x u )_yy   ( ₂

y y The difference

is the approximate loss of mass.

Chap. 9 Vector Differential Calculus. Grad, Div, Curl

Physical interpretation of the divergence

Let v be the velocity vector of the motion. We set

If we equate both expressions, divide by , and let and approach zero,

Vt

 x,y,z t v t

div u

div 





 ( ) 

0 )

( 

 

 div v

t 



or

(5) condition for the mass conservation, or continuity equationof a compressible fluid flow

If the flow is steady, then =0 and / t (6) div (v)0

If the density is constant, so that the fluid is incompressible, (7) div v = 0

* The divergence measures outflow minus inflow.

This loss of mass in B is caused by the time rate of change of the density and is equal to V t

t  



 

t z V u y u x

u  











 )

( ^{1 2 3}

where u₁(u₁)x_x(u₁)x and u₃(u₃)z_z (u₃)z

Two similar expressions are obtained by considering the other two faces. If we add these three expressions, the total mass of loss in B during the time interval Δt is approximately

Chap. 9 Vector Differential Calculus. Grad, Div, Curl

condition of the incompressibility

9.9 Curl of a Vector Field

Let

v(x, y, z) = [ v

₁

, v

₂

, v

₃

] = v

₁

i+v

₂

j+v

₃

k

be a differentiable vector function. Then the curl of the vector function is

(1) _k

y v x j v x v z i v z v y v v

v v

z y x

k j i v v

curl ( ^{3 2}) ( ^{1 3}) ( ^{2 1})

3 2 1







 







 







 











 







⇐ x, y, z are right-handed Ex. 1) Curl of a Vector Function

v = [yz, 3zx, z] = yzi + 3zxj +zk. Then

zk yj xi k

z z yj xi z

zx yz

z y x

k j i v

curl 3 (3 ) 3 2

3

















 









 

Chap. 9 Vector Differential Calculus. Grad, Div, Curl

Theorem 1

Rotating Body and Curl

The curl of the velocity field of a rotating rigid body has the direction of the axis of the rotation, and its magnitude is twice the angular speed of the rotation.

Theorem 2 Grad, Div, Curl

Gradient fields are irrotational. If a continuously differentiable vector function is the gradient of a scalar function

f

, then its curl is zero vector,

(2) curl (grad f ) = 0 and

(3) div (curl v) = 0

Theorem 3

Invariance of the Curl

Curl v is a vector. That is, it has a length and direction that are inde- pendent of the particular choice of a Cartesian coordinates system in space.

Chap. 9 Vector Differential Calculus. Grad, Div, Curl

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