Chapter 6: Q4MP (page 336)

Show that $\nabla . (U \times r) = r . (\nabla \times U)$ where U is a vector function of $x, y, z$ and $r = xi + yj + zk$ .

Short Answer

Expert verified

It has been proved that $\nabla . (U \times r) = r . (\nabla \times U)$

Step by step solution

Given Information.

It has been given that U is a vector function of x,y and z and $r = xi + yj + zk$

Definition of vector.

A quantity that has magnitude as well as direction is called a vector. It is typically denoted by an arrow in which the head determines the direction of the vector and the length determines it magnitude.

Use the formula to evaluate the expression.

Use the identity mentioned below.

$\nabla . (U \times V) = V . (\nabla \times U) - U . (\nabla \times V)$

Now let $V \to r,$

$\nabla . (U \times r) = r . (\nabla \times U) - U . (\nabla \times r)$

But r is a radial vector field, so it doesn't curl at all, hence its curl should vanish.

$\nabla . (U \times r) = r . (\nabla \times U)$

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Show that $\nabla . (U \times r) = r . (\nabla \times U)$ where U is a vector function of $x, y, z$ and $r = xi + yj + zk$ .

Short Answer

Step by step solution

Given Information.

Definition of vector.

Use the formula to evaluate the expression.

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Show that ∇.(U×r)=r.(∇×U)where U is a vector function of x,y,zandr=xi+yj+zk .

Short Answer

Step by step solution

Given Information.

Definition of vector.

Use the formula to evaluate the expression.

One App. One Place for Learning.

Most popular questions from this chapter

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Show that $\nabla . (U \times r) = r . (\nabla \times U)$ where U is a vector function of $x, y, z$ and $r = xi + yj + zk$ .