Gradient of a scalar field

Gradient of a scalar field
The vector differential operator ?, called "del", is defined in three dimensions to be
?=?/?x i+?/?y j+?/?z k
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) ?
The gradient of a 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 of f(x,y,z)=2x^2 sin?y-xy tan?z
?f/?x=4x sin?y-y tan?z, ?f/?y=2x^2 cos?y-x tan?z and ?f/?z=-xy sec^2?z
?f=(4x sin?y-y tan?z ) i+(2x^2 cos?y-x tan?z ) j-xy sec^2?z k
Divergence and the Curl 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 2: Find div F if F(x,y,z)=xzi+e^yz j-ln?(xy)k
div F=?.F=?(xz)/?x+?(e^yz )/?y-?(ln?(xy) )/?z=z+ze^yz
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 3: Find curl F if F(x,y,z)=xyi+yzj+xzk
?×F=| ?(i&j&k@?/?x&?/?y&?/?z @xy&yz&xz)|=(?(xz)/?y-?(yz)/?z)i-(?(xz)/?x-?(xy)/?z)j+(?(yz)/?x-?(xy)/?y)k
=-yi-zj-xk
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 4: Find ?f and ?^2 f for f=x^3 e^y+xy^2 z^3 at (1,0,2)
?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 (1,0,2) )=3 i+ j
?^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 (1,0,2) )=6+1=7


(1) For 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.
Find (a) ?f at (-1,2,-2) (b) ?^2 f at (1,-3,2) (c) ?.F (d) ?×F
(2) For f(x,y,z)=x^2 y-z and F(x,y,z)=xi-xyj+z^2 k. Find
(a) ?f (b) ?^2 f (c) ?.F (d) ?×F (e) ?f- ?×F