1. Verify the Identity

Let vector r = xi + yj + zk and r = |vector r|

Verify that ∇ • vector r = 3.

Must I take the partial derivative with respect to x, y, and z individually?

If so, then ∇• vector r = 1 + 1 + 1 = 3.

I do not understand why r = |vector r| is given in this problem.

2. Re: Verify the Identity

Originally Posted by USNAVY
Let vector r = xi + yj + zk and r = |vector r|
Verify that ∇ • vector r = 3.
Must I take the partial derivative with respect to x, y, and z individually?
If so, then ∇• vector r = 1 + 1 + 1 = 3.
I do not understand why r = |vector r| is given in this problem.
Do you understand that $\nabla = i\frac{\partial }{{\partial x}} + j\frac{\partial }{{\partial y}} + k\frac{\partial }{{\partial z}}$
If you know how to do dot products then : $\nabla \cdot r = x\left( {\frac{\partial }{{\partial x}}} \right) + y\left( {\frac{\partial }{{\partial y}}} \right) + z\left( {\frac{\partial }{{\partial z}}} \right) = 1 + 1 + 1$

3. Re: Verify the Identity

Originally Posted by Plato
Do you understand that $\nabla = i\frac{\partial }{{\partial x}} + j\frac{\partial }{{\partial y}} + k\frac{\partial }{{\partial z}}$
If you know how to do dot products then : $\nabla \cdot r = x\left( {\frac{\partial }{{\partial x}}} \right) + y\left( {\frac{\partial }{{\partial y}}} \right) + z\left( {\frac{\partial }{{\partial z}}} \right) = 1 + 1 + 1$
This means I was right.

4. Re: Verify the Identity

I do not understand why r = |vector r| is given in this problem.

5. Re: Verify the Identity

I suspect that there is some other part of the problem that involved $\left|\vec{r}\right|$.