Prove the Laplacian of Function g is equal to g

[zr95]

Homework Statement

Homework Equations

the gradient of g is (d/dx,d/dy,d/dz)
the divergence of g is d/dx+d/dy+d/dz

The Attempt at a Solution

When I run through even using only a few terms to see if I can get the final result of it equaling g I end up with u^2 terms as coefficients and this seems to me as if it is no longer equal.

 

[Samy_A]

Homework Statement

View attachment 96027

Homework Equations

the gradient of g is (d/dx,d/dy,d/dz)
the divergence of g is d/dx+d/dy+d/dz

The Attempt at a Solution

When I run through even using only a few terms to see if I can get the final result of it equaling g I end up with u^2 terms as coefficients and this seems to me as if it is no longer equal.

1) Can you show what you get as result?
2) What does it mean that u is a unit vector?
(3) In relevant equations, that should actually be partial derivatives.)

 

[zr95]

1) Can you show what you get as result?
2) What does it mean that u is a unit vector?

1) u1u2...un*e^(u1x1+u2x2+...unxn)
2) u is a unit vector as in it can be any vector with a magnitude of 1

 

[Samy_A]

1) u1u2...un*e^(u1x1+u2x2+...unxn)

How did you get that?
Let's start with the beginning: what is $\displaystyle \frac {\partial g}{\partial x_1}$?

 

[zr95]

How did you get that?
Let's start with the beginning: what is $\displaystyle \frac {\partial g}{\partial x_1}$

A partial derivative of g with respect to x. So we take the gradient and it becomes grad(g)=(u1e^(u1x1)+u2e^u2x2...) and so on. The thing I gave previously is after taking the divergence of the gradient.

 

[zr95]

How did you get that?
Let's start with the beginning: what is $\displaystyle \frac {\partial g}{\partial x_1}$?

Its fine now. In case anyone else needs this for future reference.