Einstein Notation and Vector Practice

Question

Prove the following identity, where i = identity tensor, x is a vector, and r = |x| :

Please use Einstein notation to solve this problem. δij is also the identity matrix.

(i∇² -∇∇)r = i/r + xx/r³

Answer 1

We want to show that: (i∇² - ∇∇)r = i/r + xx/r³

1. Calculate ∇²r

Recall that r = |x| = √(xjxj). So, ∂ir = ∂i(√(xjxj)) = xi/r. Now,

∇²r = ∇i∇ir = ∂i∂ir = ∂i(xi/r)

∂i(xi/r) = (δijr² - xixj)/r³

So,

(i∇²r) = i(δijr² - xixj)/r³

2. Calculate ∇∇r

∇∇r = ∇i∇jr = ∂i(∂j r)

We have already computed ∂ir above. Now we will compute ∂j(∂ir):

∂j(∂ir) = ∂j(xi/r) = (δijr² - xixj)/r³

So,

(∇∇r) = (δijr² - xixj)/r³

3. Combine the terms and verify the identity:

(i∇² - ∇∇)r = i(δijr² - xixj)/r³ - (δijr² - xixj)/r³ = 0

The given identity is not correct as both sides are not equal. The correct identity should be:

(i∇² - ∇∇)r = 0