Contravariant Four-gradient ESN in Wikipedia appears wrong

[SpecialEd]

Homework Statement

I am self studying relativity. In Wikipedia under the four-gradient section, the contravariant four-vector looks wrong from an Einstein summation notation point of view.

https://en.wikipedia.org/wiki/Four-vector

Homework Equations

It states:

E₀∂⁰-E₁∂¹-E₂∂²-E₃∂³ = E_α∂^α

The Attempt at a Solution

As it is it looks wrong to me. Is it wrong or is it NOT ESN or something else?

 

[TSny]

Welcome to PF!
I agree with you that it doesn't look right. For the contravariant form of the 4-gradient they have

I don't think there should be any negative signs in the first four lines. The last four lines look correct to me. So, I believe the Einstein summation works as follows $$\partial = E_\alpha \partial^\alpha = E_0 \partial ^0 + E_1 \partial ^1 +E_2 \partial ^2 + E_3 \partial ^3$$ where $$\partial^1 \equiv \frac{\partial}{\partial x_ {_1}} = -\frac{\partial}{\partial x^1}$$ etc.

 

[SpecialEd]

Welcome to PF!
I agree with you that it doesn't look right. For the contravariant form of the 4-gradient they have
View attachment 107315
I don't think there should be any negative signs in the first four lines. The last four lines look correct to me. So, I believe the Einstein summation works as follows $$\partial = E_\alpha \partial^\alpha = E_0 \partial ^0 + E_1 \partial ^1 +E_2 \partial ^2 + E_3 \partial ^3$$ where $$\partial^1 \equiv \frac{\partial}{\partial x_ {_1}} = -\frac{\partial}{\partial x^1}$$ etc.

Thanks so much for your comment and help!