Answer to Question #63508 in Calculus for hoseph ogala

Question

1 If
ϕ=2xz4−x2y
ϕ=2xz4−x2y
, find
|▽ϕ|

2 If
ϕ(x,y,z)=3x2y−y3z2
ϕ(x,y,z)=3x2y−y3z2
, find
▽ϕ
▽ϕ
at point (1,-2,-1)

3 Find a unit normal to the surface
x2y+2xz=4
x2y+2xz=4
at point (2,-2,3)
4 Let
ϕ(x,y,z)=xy2z
ϕ(x,y,z)=xy2z
and
A=xzi−xy2j+yz2k
A=xzi−xy2j+yz2k

,find
∂3∂x2∂z(ϕA)
5 Given that
ϕ=2x2y−xz3
ϕ=2x2y−xz3

find
▽2ϕ
6 If
A=xz3i−2x2yzj+2yz4
A=xz3i−2x2yzj+2yz4
, find
▽×A
▽×A

at point (1,-1,1).

7 Given that
A=A1i+A2j+A3k
A=A1i+A2j+A3k
and
r=xi+yj+zk
r=xi+yj+zk
, evaluate
▽⋅(A×r)
▽⋅(A×r)

if
▽×A=0
8 Let A=x2yi−2xzj+2yzk
A=x2yi−2xzj+2yzk

, find Curl curl A.
9 Given A=2x2i−3yzj+xz2k
A=2x2i−3yzj+xz2k
and
ϕ=2z−x3y
ϕ=2z−x3y
, find

10 Find the directional derivative of
ϕ=x2yz+4xz2
ϕ=x2yz+4xz2
at (1,-2,-1) in the direction
2i−j−2k
A⋅▽ϕ
A⋅▽ϕ
at point (1,-1,1)