Question

Prove the following vector identities letting:

V = u i + v j + w k and ∇= ∂╱∂x i + ∂╱∂y j + ∂╱∂z k that:

∇∙(fV̅) = f (∇∙V̅) + V̅ ∙∇f

Answer 1

What is fV ?

Answer 2

f is a scalar

Answer 3

v is the vector

Answer 4

I am aware. I meant write down what it actually is.

Answer 5

Explicitly

Answer 6

f (u)i + f(v) j + f(w)k

Answer 7

f is multiply u, v, and w, right?

Answer 8

yes

Answer 9

So what's the divergence of that?

Answer 10

∂╱∂x i + ∂╱∂y j + ∂╱∂z k

Answer 11

No, what is the divergence of the expression you wrote for fV ?

Answer 12

i'm not quite following what your asking me

Answer 13

\[f\vec{V} = fu\vec{i} + fv\vec{j} + fw\vec{k} \]
what is
\[\vec{\nabla} \cdot (f\vec{V}) \]
?

Answer 14

d f(u)/dx i + d f(v)/dy j + d f(w)/dz k

Answer 15

the i, j, and k are gone.

Answer 16

Okay, so expand that out.

Answer 17

so just d f(u)/dx + d f(v)/dy + d f(w)/dz

Answer 18

That's right. Now show that that equals the identity that you were given above.

Answer 19

well the right hand side of the equation is where the problems start

Answer 20

\[ \frac{\partial (fu)}{\partial x} = \frac{ \partial f}{\partial x} u + f\frac{\partial u}{\partial x}\]
etc....

Answer 21

those are equal what u just wrote above?

Answer 22

That's only the first term...

Answer 23

oh i think i just clicked what ur saying

Answer 24

okay..so the entire left side is now expanded

Answer 25

okay....i think i see where this is going now

Answer 26

if i expand the right side it's gonna come out the same way huh?

Answer 27

Yeah but it would probably be better if you just grouped the terms on the left.
\[f \frac{\partial u}{\partial x} + f\frac{\partial v}{\partial y} + f\frac{\partial w}{\partial z} = f(\vec{\nabla} \cdot \vec{V})\]
and so forth..

Answer 28

very good

Answer 29

thank you soo soo much