What themes describe the following equations?

I struggle to undestand equations 7.3,7.4.,7.5. I've watched videos about multivariable calculs  from topic derivatives of multivariable functions. And I've watched videos about partial derivatives and Laplacian operator that are presented in these equations. But I still don't understand why these Laplacian operator and gradient subtract from partial derivative. 
Could you please suggest me themes on KA to understand equations on the screenshot?