Vector Functions

Vector Functions
A vector function is a function that takes one or more variables and returns a vector.
We say that F(x,y,z) is a vector function or vector field if
F(x,y,z)=F_1 (x,y,z)i+F_2 (x,y,z)j+F_3 (x,y,z)k.
For example F(x,y,z)=x^2 yz i+2xy j+xy sec?z k
1. Gradient of a scalar field
A scalar field is a function that takes a point in space and assign a number to it , for example f(x,y,z)=x^2+cos??2y+ln?(2z+1) ?
f(1,?/6,0)=1+cos???/3+ln??1=( 3 )/( 2 )? ?
The gradient of a given scalar field f(x,y,z) is a vector field denoted by grad f or ?f and it is defined as follows :
?f=?f/?x i+?f/?y j+?f/?z k
Example 1: Find ?f for f=2x^2 sin?y-xy tan?z
Solution :
?f/?x=4x sin?y-y tan?z
?f/?y=2x^2 cos?y-x tan?z
?f/?z=-xy sec^2?z
?f=?f/?x i+?f/?y j+?f/?z k
?f=(4x sin?y-y tan?z ) i+(2x^2 cos?y-x tan?z ) j-xy sec^2?z k




2. Laplace operator
The differential operator ?^2 is called Laplace operator and it is defined as follows
?^2 f=(?^2 f)/(?x^2 )+(?^2 f)/(?y^2 )+(?^2 f)/(?z^2 )
Example 2: Find ?f and ?^2 f for f=x^3 e^y+xy^2 z^3 at (3,0,2)
Solution :
?f/?x=3x^2 e^y+y^2 z^3 ? (?^2 f)/(?x^2 )=6xe^y
?f/?y=x^3 e^y+2xyz^3 ? (?^2 f)/(?y^2 )=x^3 e^y+2xz^3
?f/?z=3xy^2 z^2 ? (?^2 f)/(?z^2 )=6xy^2 z
?f=?f/?x i+?f/?y j+?f/?z k=(3x^2 e^y+y^2 z^3 ) i+(x^3 e^y+2xyz^3 ) j+3xy^2 z^2 k
? ?f?|_( at (3,0,2) )=(3(3)^2 e^0+(0)^2 (2)^3 ) i+((3)^3 e^0+2(3)(0) (2)^3 ) j+3(3) (0)^2 (2)^2 k
? ?f?|_( at (3,0,2) )=27i+27j
?^2 f=(?^2 f)/(?x^2 )+(?^2 f)/(?y^2 )+(?^2 f)/(?z^2 )=6xe^y+x^3 e^y+2xz^3+6xy^2 z
? ?^2 f?|_(at (3,0,2) )=18+27=81
3. Divergence of a vector field
The divergence of a vector field F(x,y,z)=F_1 (x,y,z)i+F_2 (x,y,z)j+F_3 (x,y,z)k
is a scalar field div F=?.F and it is defined as follows
div F=?.F==(?F_1)/?x+(?F_2)/?y+(?F_3)/?z
Example 3: Find div F if F(x,y,z)=xzi+e^yz j-ln?(xy)k
Solution :
div F=?.F=?(xz)/?x+?(e^yz )/?y-?(ln?(xy) )/?z=z+ze^yz

4. The curl of a vector field
The curl of a vector field F(x,y,z)=F_1 (x,y,z)i+F_2 (x,y,z)j+F_3 (x,y,z)k
is another vector defined as the following determinant
curl F=?×F=| ?(i&j&k@?/?x&?/?y&?/?z @F_1&F_2&F_3 )|=((?F_3)/?y-(?F_2)/?z)i-((?F_3)/?x-(?F_1)/?z)j+((?F_2)/?x-(?F_1)/?y)k

Example 4: Find curl F if F(x,y,z)=xz sec?y i+y sin?2z j+y cos?3x k
Solution :
curl F=?×F=| ?(i&j&k@?/?x&?/?y&?/?z @F_1&F_2&F_3 )|=((?F_3)/?y-(?F_2)/?z)i-((?F_3)/?x-(?F_1)/?z)j+((?F_2)/?x-(?F_1)/?y)k
=(?(y cos?3x )/?y-?(y sin?2z )/?z)i-(?(y cos?3x )/?x-?(xz sec?y )/?z)j+(?(y sin?2z )/?x-?(xz sec?y )/?y)k
=(cos?3x-2y cos?2z )i-(-3y sin?3x-x sec?y )j+(0-xz sec?y tan?y )k
=(cos?3x-2y cos?2z )i+(3y sin?3x+x sec?y )j-(xz sec?y tan?y )k


If f(x,y,z)=x^3 y^2 z and F(x,y,z)=yze^xy i+xze^xy j+(e^xy+3 cos?3z )k.
Then find
(1) ?f at (-1,2,-2)
(2) ?^2 f at (1,-3,2)
(3) div F at (0,?6,??6)
(4) curl F