Question

Note: In the following problems, letters appear in bold face are vectors 1. show that ▽2v # (V. V) v where ▽ is the Del operator and v is any vector.

Answer 1

Del operator: - Del or nabla is an operator used in mathematics, in particular in vector calculus, as a vector differential operator. It is represented by the nabla 'nabla' nabla^2 middot v^vector = v middot (v^vector middot v^vector) Here nabla^2 v = nabla middot nabla v = div(gra v) = d^2v/dx^2 + d^2 v/dy^2 + d^2v/dz^2 we know (nabla middot nabla) v = nabla^vector = d/dx i^^+ d/dy j^^+ d/dz k^^(v^vector, v^vector) v = {(d/dx i^^+ d/dy j^^+ d/dz k^^) (d/dx i + d/dy j + d/dz k)} \r\n = (d^2/dx^2 + d^2/dy^2 + d^2/dz^2) v = d^2v/dx^2 + d^2v/dy^2 + d^2v/dz^2 V^bar^2 middot v = (v^bar, v^bar) v nabla^2 middot v is the laplacian of V.