Answer to Question #58425 in Vector Calculus for Joseph

Question

1 If
ϕ=2xz4−x2y
, find
|▽ϕ|
(√93)
(√80)
(√12)
(√110)

2 If
ϕ(x,y,z)=3x2y−y3z2
, find
▽ϕ
at point (1,-2,-1)
−12i−9j−16k
i−3j−k
2i−5j−6k
−3i−4j−2k

3 Find a unit normal to the surface
x2y+2xz=4
at point (2,-2,3)
23i−23j−23k
−15i+25j+25k
−13i+23j+23k
−17i+27j+27k

4 Let
ϕ(x,y,z)=xy2z
and
A=xzi−xy2j+yz2k
,find
∂3∂x2∂z(ϕA)
2i+2j−5k
5i−k
4i−2j
i+j

5 Given that
ϕ=2x2y−xz3
find
▽2ϕ
2y−6xz
4y−6xz
2y−xz
y+6xz

6 If
A=xz3i−2x2yzj+2yz4
, find
▽×A
at point (1,-1,1).
2j+3k
2i+j74k
i+3j+5k
3j+4k

7 Given that
A=A1i+A2j+A3k
and
r=xi+yj+zk
, evaluate
▽⋅(A×r)
if
▽×A=0
0
3
2
5

8 Let
A=x2yi−2xzj+2yzk
, find Curl curl A.
3j+4k
2x+2)k
(2x+2)j
3j−4k

9 Given
A=2x2i−3yzj+xz2k
and
ϕ=2z−x3y
, find
A⋅▽ϕ
at point (1,-1,1).
5
3
4
1

10 Find the directional derivative of
ϕ=x2yz+4xz2
at (1,-2,-1) in the direction
2i−j−2k
373
353
253
113