Question

How to show the following are true for the two vectors 𝑢̅=4𝑥𝑦𝑖̅−2𝑧𝑗̅+(11𝑥^3)𝑘̅ ⁡and 𝑣̅=(𝑥𝑦/𝑧)𝑖̅+(3𝑥^2)𝑗̅ +6𝑘̅ ? a. ∇×(𝑢̅+𝑣̅)=∇×𝑢̅+∇×𝑣̅ b. ∇.(𝑢̅+𝑣̅)=∇.𝑢̅+∇.𝑣

Answer 1

How to show the following are true for the two vectors:

`bb(ul u) = 4xy bb(ul hat i) - 2z bb(ul hat j) +11x^3 bb(ul hat k) \ \` and
`bb(ul v) = (xy)/z bb(ul hat i) +3x^2 bb(ul hat j) +6 bb(ul hat k)`?

a. `grad xx (bb(ul u) + bb(ul v)) = grad xx bb(ul u) + grad xx bb(ul v)`

b. `grad * (bb(ul u) + bb(ul v)) = grad * bb(ul u) + grad * bb(ul v)`

Explanation:

We have vectors:

`bb(ul u) = 4xy bb(ul hat i) - 2z bb(ul hat j) +11x^3 bb(ul hat k)`

`bb(ul v) = (xy)/z bb(ul hat i) +3x^2 bb(ul hat j) +6 bb(ul hat k)`

And, let us define the vector `bb(ul w) = bb(ul u) + bb(ul v)`, so that:

`bb(ul w) = bb(ul u) + bb(ul v)`

`\ \ \ = (4xy bb(ul hat i) - 2z bb(ul hat j) +11x^3 bb(ul hat k))`
`\ \ \ \ \ \ \ + ((xy)/z bb(ul hat i) +3x^2 bb(ul hat j) +6 bb(ul hat k))`

`\ \ \ = (4xy +(xy)/z) bb(ul hat i) +(3x^2- 2z) bb(ul hat j) +(11x^3+6) bb(ul hat k))`

Part (a):

Then, for the curl, we have:

`LHS = grad xx (bb(ul u) + bb(ul v))`

`\ \ \ \ \ \ \ \ = grad xx (bb(ul w))`

# \ \ \ \ \ \ \ \ = | (bb(ul hat i),bb(ul hat j),bb(ul hat k)), ((partial)/(partial x),(partial)/(partial y),(partial)/(partial z)), (4xy +(xy)/z, 3x^2- 2z, 11x^3+6) | #

# \ \ \ \ \ \ \ \ = | ((partial)/(partial y),(partial)/(partial z)), (3x^2- 2z, 11x^3+6) | bb(ul hat i) - | ((partial)/(partial x),(partial)/(partial z)), (4xy +(xy)/z, 11x^3+6) | bb(ul hat j) + | ((partial)/(partial x),(partial)/(partial y)), (4xy +(xy)/z, 3x^2- 2z) | bb(ul hat k) #

`\ \ \ \ \ \ \ \ = (0+2) bb(ul hat i) - (33x^2 + (xy)/z^2) bb(ul hat j) + (6x-4x-x/z) bb(ul hat k)`

`\ \ \ \ \ \ \ \ = 2 bb(ul hat i) - (33x^2 + (xy)/z^2) bb(ul hat j) + (2x-x/z) bb(ul hat k)`

Similarly:

`RHS = grad xx bb(ul u) + grad xx bb(ul v)`

# \ \ \ \ \ \ \ \ = | (bb(ul hat i),bb(ul hat j),bb(ul hat k)), ((partial)/(partial x),(partial)/(partial y),(partial)/(partial z)), (4xy, 3x^2, 11x^3) | + | (bb(ul hat i),bb(ul hat j),bb(ul hat k)), ((partial)/(partial x),(partial)/(partial y),(partial)/(partial z)), ((xy)/z, - 2z, 6) | #

# \ \ \ \ \ \ \ \ = | ((partial)/(partial y),(partial)/(partial z)), (3x^2, 11x^3) | bb(ul hat i) - | ((partial)/(partial x),(partial)/(partial z)), (4xy, 11x^3) | bb(ul hat j) + | ((partial)/(partial x),(partial)/(partial y)), (4xy, 3x^2) | bb(ul hat k) #
# \ \ \ \ \ \ \ \ \ \ + | ((partial)/(partial y),(partial)/(partial z)), (- 2z, +6) | bb(ul hat i) - | ((partial)/(partial x),(partial)/(partial z)), ((xy)/z, 6) | bb(ul hat j) + | ((partial)/(partial x),(partial)/(partial y)), ((xy)/z,- 2z) | bb(ul hat k) #

`\ \ \ \ \ \ \ \ = (0-0)bb(ul hat i) - (33x^2-0) bb(ul hat j) + (6x-4x) bb(ul hat k)`
`\ \ \ \ \ \ \ \ \ \ + (0+2) bb(ul hat i) - (0+(xy)/z^2) bb(ul hat j) + (0-x/z) bb(ul hat k)`

`\ \ \ \ \ \ \ \ = 2 bb(ul hat i) - (33x^2+(xy)/z^2) bb(ul hat j) + (2x-x/z) bb(ul hat k)`

`\ \ \ \ \ \ \ \ = LHS \ \ \ \` QED

Part (b):

And, for the divergence, we have:

`LHS = grad * (bb(ul u) + bb(ul v))`

`\ \ \ \ \ \ \ \ = grad * (bb(ul w))`

`\ \ \ \ \ \ \ \ = (partial)/(partial x) (4xy +(xy)/z ) + (partial)/(partial y) (3x^2- 2z) + (partial)/(partial z) (11x^3+6)`

`\ \ \ \ \ \ \ \ = (4x +(y)/z ) + (0 - 0) + (0+0)`

`\ \ \ \ \ \ \ \ = 4x +y/z`

Similarly:

`RHS = grad * bb(ul u) + grad * bb(ul v)`

`\ \ \ \ \ \ \ \ = (partial)/(partial x) (4xy) + (partial)/(partial y)(3x^2) + (partial)/(partial z)( 11x^3)`
`\ \ \ \ \ \ \ \ \ \ \ \ + (partial)/(partial x)((xy)/z) - (partial)/(partial y)(2z) + (partial)/(partial z)(6)`

`\ \ \ \ \ \ \ \ = 4x + 0 + 0 + y/z - 0 + 0`

`\ \ \ \ \ \ \ \ = 4x y/z`

`\ \ \ \ \ \ \ \ = 4x y/z`

`\ \ \ \ \ \ \ \ = LHS \ \ \ \` QED